Permutation Groups¶

class sympy.combinatorics.perm_groups.PermutationGroup(*args, dups=True, **kwargs)[source]¶

The class defining a Permutation group.

Explanation

PermutationGroup([p1, p2, ..., pn]) returns the permutation group generated by the list of permutations. This group can be supplied to Polyhedron if one desires to decorate the elements to which the indices of the permutation refer.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> from sympy.combinatorics import Polyhedron

The permutations corresponding to motion of the front, right and bottom face of a \(2 \times 2\) Rubik’s cube are defined:

>>> F = Permutation(2, 19, 21, 8)(3, 17, 20, 10)(4, 6, 7, 5)
>>> R = Permutation(1, 5, 21, 14)(3, 7, 23, 12)(8, 10, 11, 9)
>>> D = Permutation(6, 18, 14, 10)(7, 19, 15, 11)(20, 22, 23, 21)

These are passed as permutations to PermutationGroup:

>>> G = PermutationGroup(F, R, D)
>>> G.order()
3674160

The group can be supplied to a Polyhedron in order to track the objects being moved. An example involving the \(2 \times 2\) Rubik’s cube is given there, but here is a simple demonstration:

>>> a = Permutation(2, 1)
>>> b = Permutation(1, 0)
>>> G = PermutationGroup(a, b)
>>> P = Polyhedron(list('ABC'), pgroup=G)
>>> P.corners
(A, B, C)
>>> P.rotate(0) # apply permutation 0
>>> P.corners
(A, C, B)
>>> P.reset()
>>> P.corners
(A, B, C)

Or one can make a permutation as a product of selected permutations and apply them to an iterable directly:

>>> P10 = G.make_perm([0, 1])
>>> P10('ABC')
['C', 'A', 'B']

References

[R62]

Holt, D., Eick, B., O’Brien, E. “Handbook of Computational Group Theory”

[R63]

Seress, A. “Permutation Group Algorithms”

[R64]

https://en.wikipedia.org/wiki/Schreier_vector

[R65]

https://en.wikipedia.org/wiki/Nielsen_transformation#Product_replacement_algorithm

[R66]

Frank Celler, Charles R.Leedham-Green, Scott H.Murray, Alice C.Niemeyer, and E.A.O’Brien. “Generating Random Elements of a Finite Group”

[R67]

https://en.wikipedia.org/wiki/Block_%28permutation_group_theory%29

[R68]

https://algorithmist.com/wiki/Union_find

[R69]

https://en.wikipedia.org/wiki/Multiply_transitive_group#Multiply_transitive_groups

[R70]

https://en.wikipedia.org/wiki/Center_%28group_theory%29

__contains__(i)[source]¶

Return True if i is contained in PermutationGroup.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = Permutation(1, 2, 3)
>>> Permutation(3) in PermutationGroup(p)
True

__mul__(other)[source]¶

Return the direct product of two permutation groups as a permutation group.

Explanation

This implementation realizes the direct product by shifting the index set for the generators of the second group: so if we have G acting on n1 points and H acting on n2 points, G*H acts on n1 + n2 points.

Examples

>>> from sympy.combinatorics.named_groups import CyclicGroup
>>> G = CyclicGroup(5)
>>> H = G*G
>>> H
PermutationGroup([
    (9)(0 1 2 3 4),
    (5 6 7 8 9)])
>>> H.order()
25

static __new__(cls, *args, dups=True, **kwargs)[source]¶: The default constructor. Accepts Cycle and Permutation forms. Removes duplicates unless dups keyword is False.

__weakref__¶: list of weak references to the object

_coset_representative(g, H)[source]¶: Return the representative of Hg from the transversal that would be computed by self.coset_transversal(H).

classmethod _distinct_primes_lemma(primes)[source]¶: Subroutine to test if there is only one cyclic group for the order.

_eval_is_alt_sym_monte_carlo(eps=0.05, perms=None)[source]¶

A test using monte-carlo algorithm.

Parameters:

eps : float, optional

The criterion for the incorrect False return.

perms : list[Permutation], optional

If explicitly given, it tests over the given candidates for testing.

If None, it randomly computes N_eps and chooses N_eps sample of the permutation from the group.

See also

_check_cycles_alt_sym

_eval_is_alt_sym_naive(only_sym=False, only_alt=False)[source]¶: A naive test using the group order.

_p_elements_group(p)[source]¶: For an abelian p-group, return the subgroup consisting of all elements of order p (and the identity)

_random_pr_init(r, n, _random_prec_n=None)[source]¶

Initialize random generators for the product replacement algorithm.

Explanation

The implementation uses a modification of the original product replacement algorithm due to Leedham-Green, as described in [1], pp. 69-71; also, see [2], pp. 27-29 for a detailed theoretical analysis of the original product replacement algorithm, and [4].

The product replacement algorithm is used for producing random, uniformly distributed elements of a group \(G\) with a set of generators \(S\). For the initialization _random_pr_init, a list R of \(\max\{r, |S|\}\) group generators is created as the attribute G._random_gens, repeating elements of \(S\) if necessary, and the identity element of \(G\) is appended to R - we shall refer to this last element as the accumulator. Then the function random_pr() is called n times, randomizing the list R while preserving the generation of \(G\) by R. The function random_pr() itself takes two random elements g, h among all elements of R but the accumulator and replaces g with a randomly chosen element from \(\{gh, g(~h), hg, (~h)g\}\). Then the accumulator is multiplied by whatever g was replaced by. The new value of the accumulator is then returned by random_pr().

The elements returned will eventually (for n large enough) become uniformly distributed across \(G\) ([5]). For practical purposes however, the values n = 50, r = 11 are suggested in [1].

Notes

THIS FUNCTION HAS SIDE EFFECTS: it changes the attribute self._random_gens

See also

random_pr

_sylow_alt_sym(p)[source]¶

Return a p-Sylow subgroup of a symmetric or an alternating group.

Explanation

The algorithm for this is hinted at in [1], Chapter 4, Exercise 4.

For Sym(n) with n = p^i, the idea is as follows. Partition the interval [0..n-1] into p equal parts, each of length p^(i-1): [0..p^(i-1)-1], [p^(i-1)..2*p^(i-1)-1]…[(p-1)*p^(i-1)..p^i-1]. Find a p-Sylow subgroup of Sym(p^(i-1)) (treated as a subgroup of self) acting on each of the parts. Call the subgroups P_1, P_2…P_p. The generators for the subgroups P_2…P_p can be obtained from those of P_1 by applying a “shifting” permutation to them, that is, a permutation mapping [0..p^(i-1)-1] to the second part (the other parts are obtained by using the shift multiple times). The union of this permutation and the generators of P_1 is a p-Sylow subgroup of self.

For n not equal to a power of p, partition [0..n-1] in accordance with how n would be written in base p. E.g. for p=2 and n=11, 11 = 2^3 + 2^2 + 1 so the partition is [[0..7], [8..9], {10}]. To generate a p-Sylow subgroup, take the union of the generators for each of the parts. For the above example, {(0 1), (0 2)(1 3), (0 4), (1 5)(2 7)} from the first part, {(8 9)} from the second part and nothing from the third. This gives 4 generators in total, and the subgroup they generate is p-Sylow.

Alternating groups are treated the same except when p=2. In this case, (0 1)(s s+1) should be added for an appropriate s (the start of a part) for each part in the partitions.

See also

sylow_subgroup, is_alt_sym

_union_find_merge(first, second, ranks, parents, not_rep)[source]¶

Merges two classes in a union-find data structure.

Explanation

Used in the implementation of Atkinson’s algorithm as suggested in [1], pp. 83-87. The class merging process uses union by rank as an optimization. ([7])

Notes

THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives, parents, the list of class sizes, ranks, and the list of elements that are not representatives, not_rep, are changed due to class merging.

See also

minimal_block, _union_find_rep

References

[R71]

Holt, D., Eick, B., O’Brien, E. “Handbook of computational group theory”

[R77]

https://algorithmist.com/wiki/Union_find

_union_find_rep(num, parents)[source]¶

Find representative of a class in a union-find data structure.

Explanation

Used in the implementation of Atkinson’s algorithm as suggested in [1], pp. 83-87. After the representative of the class to which num belongs is found, path compression is performed as an optimization ([7]).

Notes

THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives, parents, is altered due to path compression.

See also

minimal_block, _union_find_merge

References

[R73]

Holt, D., Eick, B., O’Brien, E. “Handbook of computational group theory”

[R79]

https://algorithmist.com/wiki/Union_find

_verify(K, phi, z, alpha)[source]¶

Return a list of relators rels in generators gens`_h` that are mapped to ``H.generators by phi so that given a finite presentation <gens_k | rels_k> of K on a subset of gens_h <gens_h | rels_k + rels> is a finite presentation of H.

Explanation

H should be generated by the union of K.generators and z (a single generator), and H.stabilizer(alpha) == K; phi is a canonical injection from a free group into a permutation group containing H.

The algorithm is described in [1], Chapter 6.

Examples

>>> from sympy.combinatorics import free_group, Permutation, PermutationGroup
>>> from sympy.combinatorics.homomorphisms import homomorphism
>>> from sympy.combinatorics.fp_groups import FpGroup

>>> H = PermutationGroup(Permutation(0, 2), Permutation (1, 5))
>>> K = PermutationGroup(Permutation(5)(0, 2))
>>> F = free_group("x_0 x_1")[0]
>>> gens = F.generators
>>> phi = homomorphism(F, H, F.generators, H.generators)
>>> rels_k = [gens[0]**2] # relators for presentation of K
>>> z= Permutation(1, 5)
>>> check, rels_h = H._verify(K, phi, z, 1)
>>> check
True
>>> rels = rels_k + rels_h
>>> G = FpGroup(F, rels) # presentation of H
>>> G.order() == H.order()
True

abelian_invariants()[source]¶

Returns the abelian invariants for the given group. Let G be a nontrivial finite abelian group. Then G is isomorphic to the direct product of finitely many nontrivial cyclic groups of prime-power order.

Explanation

The prime-powers that occur as the orders of the factors are uniquely determined by G. More precisely, the primes that occur in the orders of the factors in any such decomposition of G are exactly the primes that divide |G| and for any such prime p, if the orders of the factors that are p-groups in one such decomposition of G are p^{t_1} >= p^{t_2} >= ... p^{t_r}, then the orders of the factors that are p-groups in any such decomposition of G are p^{t_1} >= p^{t_2} >= ... p^{t_r}.

The uniquely determined integers p^{t_1} >= p^{t_2} >= ... p^{t_r}, taken for all primes that divide |G| are called the invariants of the nontrivial group G as suggested in ([14], p. 542).

Notes

We adopt the convention that the invariants of a trivial group are [].

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.abelian_invariants()
[2]
>>> from sympy.combinatorics import CyclicGroup
>>> G = CyclicGroup(7)
>>> G.abelian_invariants()
[7]

property base¶

Return a base from the Schreier-Sims algorithm.

Explanation

For a permutation group \(G\), a base is a sequence of points \(B = (b_1, b_2, \dots, b_k)\) such that no element of \(G\) apart from the identity fixes all the points in \(B\). The concepts of a base and strong generating set and their applications are discussed in depth in [1], pp. 87-89 and [2], pp. 55-57.

An alternative way to think of \(B\) is that it gives the indices of the stabilizer cosets that contain more than the identity permutation.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> G = PermutationGroup([Permutation(0, 1, 3)(2, 4)])
>>> G.base
[0, 2]

baseswap(base, strong_gens, pos, randomized=False, transversals=None, basic_orbits=None, strong_gens_distr=None)[source]¶

Swap two consecutive base points in base and strong generating set.

Parameters:

base, strong_gens

The base and strong generating set.

pos

The position at which swapping is performed.

randomized

A switch between randomized and deterministic version.

transversals

The transversals for the basic orbits, if known.

basic_orbits

The basic orbits, if known.

strong_gens_distr

The strong generators distributed by basic stabilizers, if known.

Returns:

(base, strong_gens)

base is the new base, and strong_gens is a generating set relative to it.

Explanation

If a base for a group \(G\) is given by \((b_1, b_2, \dots, b_k)\), this function returns a base \((b_1, b_2, \dots, b_{i+1}, b_i, \dots, b_k)\), where \(i\) is given by pos, and a strong generating set relative to that base. The original base and strong generating set are not modified.

The randomized version (default) is of Las Vegas type.

Examples

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> S = SymmetricGroup(4)
>>> S.schreier_sims()
>>> S.base
[0, 1, 2]
>>> base, gens = S.baseswap(S.base, S.strong_gens, 1, randomized=False)
>>> base, gens
([0, 2, 1],
[(0 1 2 3), (3)(0 1), (1 3 2),
 (2 3), (1 3)])

check that base, gens is a BSGS

>>> S1 = PermutationGroup(gens)
>>> _verify_bsgs(S1, base, gens)
True

Notes

The deterministic version of the algorithm is discussed in [1], pp. 102-103; the randomized version is discussed in [1], p.103, and [2], p.98. It is of Las Vegas type. Notice that [1] contains a mistake in the pseudocode and discussion of BASESWAP: on line 3 of the pseudocode, \(|\beta_{i+1}^{\left\langle T\right\rangle}|\) should be replaced by \(|\beta_{i}^{\left\langle T\right\rangle}|\), and the same for the discussion of the algorithm.

See also

schreier_sims

property basic_orbits¶

Return the basic orbits relative to a base and strong generating set.

Explanation

If \((b_1, b_2, \dots, b_k)\) is a base for a group \(G\), and \(G^{(i)} = G_{b_1, b_2, \dots, b_{i-1}}\) is the i-th basic stabilizer (so that \(G^{(1)} = G\)), the i-th basic orbit relative to this base is the orbit of \(b_i\) under \(G^{(i)}\). See [1], pp. 87-89 for more information.

Examples

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(4)
>>> S.basic_orbits
[[0, 1, 2, 3], [1, 2, 3], [2, 3]]

property basic_stabilizers¶

Return a chain of stabilizers relative to a base and strong generating set.

Explanation

The i-th basic stabilizer \(G^{(i)}\) relative to a base \((b_1, b_2, \dots, b_k)\) is \(G_{b_1, b_2, \dots, b_{i-1}}\). For more information, see [1], pp. 87-89.

Examples

>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> A = AlternatingGroup(4)
>>> A.schreier_sims()
>>> A.base
[0, 1]
>>> for g in A.basic_stabilizers:
...     print(g)
...
PermutationGroup([
    (3)(0 1 2),
    (1 2 3)])
PermutationGroup([
    (1 2 3)])

property basic_transversals¶

Return basic transversals relative to a base and strong generating set.

Explanation

The basic transversals are transversals of the basic orbits. They are provided as a list of dictionaries, each dictionary having keys - the elements of one of the basic orbits, and values - the corresponding transversal elements. See [1], pp. 87-89 for more information.

Examples

>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> A = AlternatingGroup(4)
>>> A.basic_transversals
[{0: (3), 1: (3)(0 1 2), 2: (3)(0 2 1), 3: (0 3 1)}, {1: (3), 2: (1 2 3), 3: (1 3 2)}]

center()[source]¶

Return the center of a permutation group.

Explanation

The center for a group \(G\) is defined as \(Z(G) = \{z\in G | \forall g\in G, zg = gz \}\), the set of elements of \(G\) that commute with all elements of \(G\). It is equal to the centralizer of \(G\) inside \(G\), and is naturally a subgroup of \(G\) ([9]).

Examples

>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(4)
>>> G = D.center()
>>> G.order()
2

Notes

This is a naive implementation that is a straightforward application of .centralizer()

See also

centralizer

centralizer(other)[source]¶

Return the centralizer of a group/set/element.

Parameters:: other

a permutation group/list of permutations/single permutation

Explanation

The centralizer of a set of permutations S inside a group G is the set of elements of G that commute with all elements of S:

`C_G(S) = \{ g \in G | gs = sg \forall s \in S\}` ([10])

Usually, S is a subset of G, but if G is a proper subgroup of the full symmetric group, we allow for S to have elements outside G.

It is naturally a subgroup of G; the centralizer of a permutation group is equal to the centralizer of any set of generators for that group, since any element commuting with the generators commutes with any product of the generators.

Examples

>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup)
>>> S = SymmetricGroup(6)
>>> C = CyclicGroup(6)
>>> H = S.centralizer(C)
>>> H.is_subgroup(C)
True

Notes

The implementation is an application of .subgroup_search() with tests using a specific base for the group G.

See also

subgroup_search

commutator(G, H)[source]¶

Return the commutator of two subgroups.

Explanation

For a permutation group K and subgroups G, H, the commutator of G and H is defined as the group generated by all the commutators \([g, h] = hgh^{-1}g^{-1}\) for g in G and h in H. It is naturally a subgroup of K ([1], p.27).

Examples

>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup)
>>> S = SymmetricGroup(5)
>>> A = AlternatingGroup(5)
>>> G = S.commutator(S, A)
>>> G.is_subgroup(A)
True

Notes

The commutator of two subgroups \(H, G\) is equal to the normal closure of the commutators of all the generators, i.e. \(hgh^{-1}g^{-1}\) for \(h\) a generator of \(H\) and \(g\) a generator of \(G\) ([1], p.28)

See also

derived_subgroup

composition_series()[source]¶

Return the composition series for a group as a list of permutation groups.

Explanation

The composition series for a group \(G\) is defined as a subnormal series \(G = H_0 > H_1 > H_2 \ldots\) A composition series is a subnormal series such that each factor group \(H(i+1) / H(i)\) is simple. A subnormal series is a composition series only if it is of maximum length.

The algorithm works as follows: Starting with the derived series the idea is to fill the gap between \(G = der[i]\) and \(H = der[i+1]\) for each \(i\) independently. Since, all subgroups of the abelian group \(G/H\) are normal so, first step is to take the generators \(g\) of \(G\) and add them to generators of \(H\) one by one.

The factor groups formed are not simple in general. Each group is obtained from the previous one by adding one generator \(g\), if the previous group is denoted by \(H\) then the next group \(K\) is generated by \(g\) and \(H\). The factor group \(K/H\) is cyclic and it’s order is \(K.order()//G.order()\). The series is then extended between \(K\) and \(H\) by groups generated by powers of \(g\) and \(H\). The series formed is then prepended to the already existing series.

Examples

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.named_groups import CyclicGroup
>>> S = SymmetricGroup(12)
>>> G = S.sylow_subgroup(2)
>>> C = G.composition_series()
>>> [H.order() for H in C]
[1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1]
>>> G = S.sylow_subgroup(3)
>>> C = G.composition_series()
>>> [H.order() for H in C]
[243, 81, 27, 9, 3, 1]
>>> G = CyclicGroup(12)
>>> C = G.composition_series()
>>> [H.order() for H in C]
[12, 6, 3, 1]

conjugacy_class(x)[source]¶

Return the conjugacy class of an element in the group.

Explanation

The conjugacy class of an element g in a group G is the set of elements x in G that are conjugate with g, i.e. for which

g = xax^{-1}

for some a in G.

Note that conjugacy is an equivalence relation, and therefore that conjugacy classes are partitions of G. For a list of all the conjugacy classes of the group, use the conjugacy_classes() method.

In a permutation group, each conjugacy class corresponds to a particular \(cycle structure': for example, in ``S_3`\), the conjugacy classes are:

the identity class, {()}

all transpositions, {(1 2), (1 3), (2 3)}

all 3-cycles, {(1 2 3), (1 3 2)}

Examples

>>> from sympy.combinatorics import Permutation, SymmetricGroup
>>> S3 = SymmetricGroup(3)
>>> S3.conjugacy_class(Permutation(0, 1, 2))
{(0 1 2), (0 2 1)}

Notes

This procedure computes the conjugacy class directly by finding the orbit of the element under conjugation in G. This algorithm is only feasible for permutation groups of relatively small order, but is like the orbit() function itself in that respect.

conjugacy_classes()[source]¶

Return the conjugacy classes of the group.

Explanation

As described in the documentation for the .conjugacy_class() function, conjugacy is an equivalence relation on a group G which partitions the set of elements. This method returns a list of all these conjugacy classes of G.

Examples

>>> from sympy.combinatorics import SymmetricGroup
>>> SymmetricGroup(3).conjugacy_classes()
[{(2)}, {(0 1 2), (0 2 1)}, {(0 2), (1 2), (2)(0 1)}]

contains(g, strict=True)[source]¶

Test if permutation g belong to self, G.

Explanation

If g is an element of G it can be written as a product of factors drawn from the cosets of G’s stabilizers. To see if g is one of the actual generators defining the group use G.has(g).

If strict is not True, g will be resized, if necessary, to match the size of permutations in self.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup

>>> a = Permutation(1, 2)
>>> b = Permutation(2, 3, 1)
>>> G = PermutationGroup(a, b, degree=5)
>>> G.contains(G[0]) # trivial check
True
>>> elem = Permutation([[2, 3]], size=5)
>>> G.contains(elem)
True
>>> G.contains(Permutation(4)(0, 1, 2, 3))
False

If strict is False, a permutation will be resized, if necessary:

>>> H = PermutationGroup(Permutation(5))
>>> H.contains(Permutation(3))
False
>>> H.contains(Permutation(3), strict=False)
True

To test if a given permutation is present in the group:

>>> elem in G.generators
False
>>> G.has(elem)
False

coset_factor(g, factor_index=False)[source]¶

Return G’s (self’s) coset factorization of g

Explanation

If g is an element of G then it can be written as the product of permutations drawn from the Schreier-Sims coset decomposition,

The permutations returned in f are those for which the product gives g: g = f[n]*...f[1]*f[0] where n = len(B) and B = G.base. f[i] is one of the permutations in self._basic_orbits[i].

If factor_index==True, returns a tuple [b[0],..,b[n]], where b[i] belongs to self._basic_orbits[i]

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(0, 1, 3, 7, 6, 4)(2, 5)
>>> b = Permutation(0, 1, 3, 2)(4, 5, 7, 6)
>>> G = PermutationGroup([a, b])

Define g:

>>> g = Permutation(7)(1, 2, 4)(3, 6, 5)

Confirm that it is an element of G:

>>> G.contains(g)
True

Thus, it can be written as a product of factors (up to 3) drawn from u. See below that a factor from u1 and u2 and the Identity permutation have been used:

>>> f = G.coset_factor(g)
>>> f[2]*f[1]*f[0] == g
True
>>> f1 = G.coset_factor(g, True); f1
[0, 4, 4]
>>> tr = G.basic_transversals
>>> f[0] == tr[0][f1[0]]
True

If g is not an element of G then [] is returned:

>>> c = Permutation(5, 6, 7)
>>> G.coset_factor(c)
[]

See also

sympy.combinatorics.util._strip

coset_rank(g)[source]¶

rank using Schreier-Sims representation.

Explanation

The coset rank of g is the ordering number in which it appears in the lexicographic listing according to the coset decomposition

The ordering is the same as in G.generate(method=’coset’). If g does not belong to the group it returns None.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(0, 1, 3, 7, 6, 4)(2, 5)
>>> b = Permutation(0, 1, 3, 2)(4, 5, 7, 6)
>>> G = PermutationGroup([a, b])
>>> c = Permutation(7)(2, 4)(3, 5)
>>> G.coset_rank(c)
16
>>> G.coset_unrank(16)
(7)(2 4)(3 5)

See also

coset_factor

coset_table(H)[source]¶: Return the standardised (right) coset table of self in H as a list of lists.

coset_transversal(H)[source]¶: Return a transversal of the right cosets of self by its subgroup H using the second method described in [1], Subsection 4.6.7

coset_unrank(rank, af=False)[source]¶

unrank using Schreier-Sims representation

coset_unrank is the inverse operation of coset_rank if 0 <= rank < order; otherwise it returns None.

property degree¶

Returns the size of the permutations in the group.

Explanation

The number of permutations comprising the group is given by len(group); the number of permutations that can be generated by the group is given by group.order().

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 0, 2])
>>> G = PermutationGroup([a])
>>> G.degree
3
>>> len(G)
1
>>> G.order()
2
>>> list(G.generate())
[(2), (2)(0 1)]

See also

order

derived_series()[source]¶

Return the derived series for the group.

Returns:

A list of permutation groups containing the members of the derived

series in the order \(G = G_0, G_1, G_2, \ldots\).

Explanation

The derived series for a group \(G\) is defined as \(G = G_0 > G_1 > G_2 > \ldots\) where \(G_i = [G_{i-1}, G_{i-1}]\), i.e. \(G_i\) is the derived subgroup of \(G_{i-1}\), for \(i\in\mathbb{N}\). When we have \(G_k = G_{k-1}\) for some \(k\in\mathbb{N}\), the series terminates.

Examples

>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup, DihedralGroup)
>>> A = AlternatingGroup(5)
>>> len(A.derived_series())
1
>>> S = SymmetricGroup(4)
>>> len(S.derived_series())
4
>>> S.derived_series()[1].is_subgroup(AlternatingGroup(4))
True
>>> S.derived_series()[2].is_subgroup(DihedralGroup(2))
True

See also

derived_subgroup

derived_subgroup()[source]¶

Compute the derived subgroup.

Explanation

The derived subgroup, or commutator subgroup is the subgroup generated by all commutators \([g, h] = hgh^{-1}g^{-1}\) for \(g, h\in G\) ; it is equal to the normal closure of the set of commutators of the generators ([1], p.28, [11]).

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 0, 2, 4, 3])
>>> b = Permutation([0, 1, 3, 2, 4])
>>> G = PermutationGroup([a, b])
>>> C = G.derived_subgroup()
>>> list(C.generate(af=True))
[[0, 1, 2, 3, 4], [0, 1, 3, 4, 2], [0, 1, 4, 2, 3]]

See also

derived_series

property elements¶

Returns all the elements of the permutation group as a list

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = PermutationGroup(Permutation(1, 3), Permutation(1, 2))
>>> p.elements
[(3), (3)(1 2), (1 3), (2 3), (1 2 3), (1 3 2)]

equals(other)[source]¶

Return True if PermutationGroup generated by elements in the group are same i.e they represent the same PermutationGroup.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = Permutation(0, 1, 2, 3, 4, 5)
>>> G = PermutationGroup([p, p**2])
>>> H = PermutationGroup([p**2, p])
>>> G.generators == H.generators
False
>>> G.equals(H)
True

generate(method='coset', af=False)[source]¶

Return iterator to generate the elements of the group.

Explanation

Iteration is done with one of these methods:

method='coset'  using the Schreier-Sims coset representation
method='dimino' using the Dimino method

If af = True it yields the array form of the permutations

Examples

>>> from sympy.combinatorics import PermutationGroup
>>> from sympy.combinatorics.polyhedron import tetrahedron

The permutation group given in the tetrahedron object is also true groups:

>>> G = tetrahedron.pgroup
>>> G.is_group
True

Also the group generated by the permutations in the tetrahedron pgroup – even the first two – is a proper group:

>>> H = PermutationGroup(G[0], G[1])
>>> J = PermutationGroup(list(H.generate())); J
PermutationGroup([
    (0 1)(2 3),
    (1 2 3),
    (1 3 2),
    (0 3 1),
    (0 2 3),
    (0 3)(1 2),
    (0 1 3),
    (3)(0 2 1),
    (0 3 2),
    (3)(0 1 2),
    (0 2)(1 3)])
>>> _.is_group
True

generate_dimino(af=False)[source]¶

Yield group elements using Dimino’s algorithm.

If af == True it yields the array form of the permutations.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_dimino(af=True))
[[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1],
 [0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]]

References

[R75]

The Implementation of Various Algorithms for Permutation Groups in the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis

generate_schreier_sims(af=False)[source]¶

Yield group elements using the Schreier-Sims representation in coset_rank order

If af = True it yields the array form of the permutations

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_schreier_sims(af=True))
[[0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 2, 1],
 [0, 1, 3, 2], [0, 2, 3, 1], [0, 3, 1, 2]]

generator_product(g, original=False)[source]¶: Return a list of strong generators \([s1, \dots, sn]\) s.t \(g = sn \times \dots \times s1\). If original=True, make the list contain only the original group generators

property generators¶

Returns the generators of the group.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.generators
[(1 2), (2)(0 1)]

property identity¶: Return the identity element of the permutation group.

index(H)[source]¶

Returns the index of a permutation group.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(1,2,3)
>>> b =Permutation(3)
>>> G = PermutationGroup([a])
>>> H = PermutationGroup([b])
>>> G.index(H)
3

property is_abelian¶

Test if the group is Abelian.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.is_abelian
False
>>> a = Permutation([0, 2, 1])
>>> G = PermutationGroup([a])
>>> G.is_abelian
True

is_alt_sym(eps=0.05, _random_prec=None)[source]¶

Monte Carlo test for the symmetric/alternating group for degrees >= 8.

Explanation

More specifically, it is one-sided Monte Carlo with the answer True (i.e., G is symmetric/alternating) guaranteed to be correct, and the answer False being incorrect with probability eps.

For degree < 8, the order of the group is checked so the test is deterministic.

Notes

The algorithm itself uses some nontrivial results from group theory and number theory: 1) If a transitive group G of degree n contains an element with a cycle of length n/2 < p < n-2 for p a prime, G is the symmetric or alternating group ([1], pp. 81-82) 2) The proportion of elements in the symmetric/alternating group having the property described in 1) is approximately \(\log(2)/\log(n)\) ([1], p.82; [2], pp. 226-227). The helper function _check_cycles_alt_sym is used to go over the cycles in a permutation and look for ones satisfying 1).

Examples

>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(10)
>>> D.is_alt_sym()
False

See also

_check_cycles_alt_sym

property is_alternating¶

Return True if the group is alternating.

Examples

>>> from sympy.combinatorics import AlternatingGroup
>>> g = AlternatingGroup(5)
>>> g.is_alternating
True

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> g = PermutationGroup(
...     Permutation(0, 1, 2, 3, 4),
...     Permutation(2, 3, 4))
>>> g.is_alternating
True

Notes

This uses a naive test involving the computation of the full group order. If you need more quicker taxonomy for large groups, you can use PermutationGroup.is_alt_sym(). However, PermutationGroup.is_alt_sym() may not be accurate and is not able to distinguish between an alternating group and a symmetric group.

See also

is_alt_sym

property is_cyclic¶

Return True if the group is Cyclic.

Examples

>>> from sympy.combinatorics.named_groups import AbelianGroup
>>> G = AbelianGroup(3, 4)
>>> G.is_cyclic
True
>>> G = AbelianGroup(4, 4)
>>> G.is_cyclic
False

Notes

If the order of a group \(n\) can be factored into the distinct primes \(p_1, p_2, \dots , p_s\) and if

\[\forall i, j \in \{1, 2, \dots, s \}: p_i \not \equiv 1 \pmod {p_j}\]

holds true, there is only one group of the order \(n\) which is a cyclic group [R76]. This is a generalization of the lemma that the group of order \(15, 35, \dots\) are cyclic.

And also, these additional lemmas can be used to test if a group is cyclic if the order of the group is already found.

If the group is abelian and the order of the group is square-free, the group is cyclic.
If the order of the group is less than \(6\) and is not \(4\), the group is cyclic.
If the order of the group is prime, the group is cyclic.

References

[R76] (1,2)

1978: John S. Rose: A Course on Group Theory, Introduction to Finite Group Theory: 1.4

property is_dihedral¶

Return True if the group is dihedral.

Examples

>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.named_groups import SymmetricGroup, CyclicGroup
>>> G = PermutationGroup(Permutation(1, 6)(2, 5)(3, 4), Permutation(0, 1, 2, 3, 4, 5, 6))
>>> G.is_dihedral
True
>>> G = SymmetricGroup(3)
>>> G.is_dihedral
True
>>> G = CyclicGroup(6)
>>> G.is_dihedral
False

References

[Di1]

https://math.stackexchange.com/questions/827230/given-a-cayley-table-is-there-an-algorithm-to-determine-if-it-is-a-dihedral-gro/827273#827273

[Di2]

https://kconrad.math.uconn.edu/blurbs/grouptheory/dihedral.pdf

[Di3]

https://kconrad.math.uconn.edu/blurbs/grouptheory/dihedral2.pdf

[Di4]

https://en.wikipedia.org/wiki/Dihedral_group

is_elementary(p)[source]¶

Return True if the group is elementary abelian. An elementary abelian group is a finite abelian group, where every nontrivial element has order \(p\), where \(p\) is a prime.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> G = PermutationGroup([a])
>>> G.is_elementary(2)
True
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([3, 1, 2, 0])
>>> G = PermutationGroup([a, b])
>>> G.is_elementary(2)
True
>>> G.is_elementary(3)
False

property is_nilpotent¶

Test if the group is nilpotent.

Explanation

A group \(G\) is nilpotent if it has a central series of finite length. Alternatively, \(G\) is nilpotent if its lower central series terminates with the trivial group. Every nilpotent group is also solvable ([1], p.29, [12]).

Examples

>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup)
>>> C = CyclicGroup(6)
>>> C.is_nilpotent
True
>>> S = SymmetricGroup(5)
>>> S.is_nilpotent
False

See also

lower_central_series, is_solvable

is_normal(gr, strict=True)[source]¶

Test if G=self is a normal subgroup of gr.

Explanation

G is normal in gr if for each g2 in G, g1 in gr, g = g1*g2*g1**-1 belongs to G It is sufficient to check this for each g1 in gr.generators and g2 in G.generators.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 2, 0])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G1 = PermutationGroup([a, Permutation([2, 0, 1])])
>>> G1.is_normal(G)
True

property is_perfect¶

Return True if the group is perfect. A group is perfect if it equals to its derived subgroup.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(1,2,3)(4,5)
>>> b = Permutation(1,2,3,4,5)
>>> G = PermutationGroup([a, b])
>>> G.is_perfect
False

property is_polycyclic¶

Return True if a group is polycyclic. A group is polycyclic if it has a subnormal series with cyclic factors. For finite groups, this is the same as if the group is solvable.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([2, 0, 1, 3])
>>> G = PermutationGroup([a, b])
>>> G.is_polycyclic
True

is_primitive(randomized=True)[source]¶

Test if a group is primitive.

Explanation

A permutation group G acting on a set S is called primitive if S contains no nontrivial block under the action of G (a block is nontrivial if its cardinality is more than 1).

Notes

The algorithm is described in [1], p.83, and uses the function minimal_block to search for blocks of the form \(\{0, k\}\) for k ranging over representatives for the orbits of \(G_0\), the stabilizer of 0. This algorithm has complexity \(O(n^2)\) where n is the degree of the group, and will perform badly if \(G_0\) is small.

There are two implementations offered: one finds \(G_0\) deterministically using the function stabilizer, and the other (default) produces random elements of \(G_0\) using random_stab, hoping that they generate a subgroup of \(G_0\) with not too many more orbits than \(G_0\) (this is suggested in [1], p.83). Behavior is changed by the randomized flag.

Examples

>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(10)
>>> D.is_primitive()
False

See also

minimal_block, random_stab

property is_solvable¶

Test if the group is solvable.

G is solvable if its derived series terminates with the trivial group ([1], p.29).

Examples

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(3)
>>> S.is_solvable
True

See also

is_nilpotent, derived_series

is_subgroup(G, strict=True)[source]¶

Return True if all elements of self belong to G.

If strict is False then if self’s degree is smaller than G’s, the elements will be resized to have the same degree.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> from sympy.combinatorics import SymmetricGroup, CyclicGroup

Testing is strict by default: the degree of each group must be the same:

>>> p = Permutation(0, 1, 2, 3, 4, 5)
>>> G1 = PermutationGroup([Permutation(0, 1, 2), Permutation(0, 1)])
>>> G2 = PermutationGroup([Permutation(0, 2), Permutation(0, 1, 2)])
>>> G3 = PermutationGroup([p, p**2])
>>> assert G1.order() == G2.order() == G3.order() == 6
>>> G1.is_subgroup(G2)
True
>>> G1.is_subgroup(G3)
False
>>> G3.is_subgroup(PermutationGroup(G3[1]))
False
>>> G3.is_subgroup(PermutationGroup(G3[0]))
True

To ignore the size, set strict to False:

>>> S3 = SymmetricGroup(3)
>>> S5 = SymmetricGroup(5)
>>> S3.is_subgroup(S5, strict=False)
True
>>> C7 = CyclicGroup(7)
>>> G = S5*C7
>>> S5.is_subgroup(G, False)
True
>>> C7.is_subgroup(G, 0)
False

property is_symmetric¶

Return True if the group is symmetric.

Examples

>>> from sympy.combinatorics import SymmetricGroup
>>> g = SymmetricGroup(5)
>>> g.is_symmetric
True

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> g = PermutationGroup(
...     Permutation(0, 1, 2, 3, 4),
...     Permutation(2, 3))
>>> g.is_symmetric
True

Notes

See also

is_alt_sym

is_transitive(strict=True)[source]¶

Test if the group is transitive.

Explanation

A group is transitive if it has a single orbit.

If strict is False the group is transitive if it has a single orbit of length different from 1.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([2, 0, 1, 3])
>>> G1 = PermutationGroup([a, b])
>>> G1.is_transitive()
False
>>> G1.is_transitive(strict=False)
True
>>> c = Permutation([2, 3, 0, 1])
>>> G2 = PermutationGroup([a, c])
>>> G2.is_transitive()
True
>>> d = Permutation([1, 0, 2, 3])
>>> e = Permutation([0, 1, 3, 2])
>>> G3 = PermutationGroup([d, e])
>>> G3.is_transitive() or G3.is_transitive(strict=False)
False

property is_trivial¶

Test if the group is the trivial group.

This is true if the group contains only the identity permutation.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> G = PermutationGroup([Permutation([0, 1, 2])])
>>> G.is_trivial
True

lower_central_series()[source]¶

Return the lower central series for the group.

The lower central series for a group \(G\) is the series \(G = G_0 > G_1 > G_2 > \ldots\) where \(G_k = [G, G_{k-1}]\), i.e. every term after the first is equal to the commutator of \(G\) and the previous term in \(G1\) ([1], p.29).

Returns:: A list of permutation groups in the order \(G = G_0, G_1, G_2, \ldots\)

Examples

>>> from sympy.combinatorics.named_groups import (AlternatingGroup,
... DihedralGroup)
>>> A = AlternatingGroup(4)
>>> len(A.lower_central_series())
2
>>> A.lower_central_series()[1].is_subgroup(DihedralGroup(2))
True

See also

commutator, derived_series

make_perm(n, seed=None)[source]¶

Multiply n randomly selected permutations from pgroup together, starting with the identity permutation. If n is a list of integers, those integers will be used to select the permutations and they will be applied in L to R order: make_perm((A, B, C)) will give CBA(I) where I is the identity permutation.

seed is used to set the seed for the random selection of permutations from pgroup. If this is a list of integers, the corresponding permutations from pgroup will be selected in the order give. This is mainly used for testing purposes.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a, b = [Permutation([1, 0, 3, 2]), Permutation([1, 3, 0, 2])]
>>> G = PermutationGroup([a, b])
>>> G.make_perm(1, [0])
(0 1)(2 3)
>>> G.make_perm(3, [0, 1, 0])
(0 2 3 1)
>>> G.make_perm([0, 1, 0])
(0 2 3 1)

See also

random

property max_div¶

Maximum proper divisor of the degree of a permutation group.

Explanation

Obviously, this is the degree divided by its minimal proper divisor (larger than 1, if one exists). As it is guaranteed to be prime, the sieve from sympy.ntheory is used. This function is also used as an optimization tool for the functions minimal_block and _union_find_merge.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> G = PermutationGroup([Permutation([0, 2, 1, 3])])
>>> G.max_div
2

See also

minimal_block, _union_find_merge

minimal_block(points)[source]¶

For a transitive group, finds the block system generated by points.

Explanation

If a group G acts on a set S, a nonempty subset B of S is called a block under the action of G if for all g in G we have gB = B (g fixes B) or gB and B have no common points (g moves B entirely). ([1], p.23; [6]).

The distinct translates gB of a block B for g in G partition the set S and this set of translates is known as a block system. Moreover, we obviously have that all blocks in the partition have the same size, hence the block size divides |S| ([1], p.23). A G-congruence is an equivalence relation ~ on the set S such that a ~ b implies g(a) ~ g(b) for all g in G. For a transitive group, the equivalence classes of a G-congruence and the blocks of a block system are the same thing ([1], p.23).

The algorithm below checks the group for transitivity, and then finds the G-congruence generated by the pairs (p_0, p_1), (p_0, p_2), ..., (p_0,p_{k-1}) which is the same as finding the maximal block system (i.e., the one with minimum block size) such that p_0, ..., p_{k-1} are in the same block ([1], p.83).

It is an implementation of Atkinson’s algorithm, as suggested in [1], and manipulates an equivalence relation on the set S using a union-find data structure. The running time is just above \(O(|points||S|)\). ([1], pp. 83-87; [7]).

Examples

>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(10)
>>> D.minimal_block([0, 5])
[0, 1, 2, 3, 4, 0, 1, 2, 3, 4]
>>> D.minimal_block([0, 1])
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

minimal_blocks(randomized=True)[source]¶

For a transitive group, return the list of all minimal block systems. If a group is intransitive, return \(False\).

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> DihedralGroup(6).minimal_blocks()
[[0, 1, 0, 1, 0, 1], [0, 1, 2, 0, 1, 2]]
>>> G = PermutationGroup(Permutation(1,2,5))
>>> G.minimal_blocks()
False

normal_closure(other, k=10)[source]¶

Return the normal closure of a subgroup/set of permutations.

Parameters:

other

a subgroup/list of permutations/single permutation

an implementation-specific parameter that determines the number of conjugates that are adjoined to other at once

Explanation

If S is a subset of a group G, the normal closure of A in G is defined as the intersection of all normal subgroups of G that contain A ([1], p.14). Alternatively, it is the group generated by the conjugates x^{-1}yx for x a generator of G and y a generator of the subgroup \left\langle S\right\rangle generated by S (for some chosen generating set for \left\langle S\right\rangle) ([1], p.73).

Examples

>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup, AlternatingGroup)
>>> S = SymmetricGroup(5)
>>> C = CyclicGroup(5)
>>> G = S.normal_closure(C)
>>> G.order()
60
>>> G.is_subgroup(AlternatingGroup(5))
True

Notes

The algorithm is described in [1], pp. 73-74; it makes use of the generation of random elements for permutation groups by the product replacement algorithm.

orbit(alpha, action='tuples')[source]¶

Compute the orbit of alpha \(\{g(\alpha) | g \in G\}\) as a set.

Explanation

The time complexity of the algorithm used here is \(O(|Orb|*r)\) where \(|Orb|\) is the size of the orbit and r is the number of generators of the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21. Here alpha can be a single point, or a list of points.

If alpha is a single point, the ordinary orbit is computed. if alpha is a list of points, there are three available options:

‘union’ - computes the union of the orbits of the points in the list ‘tuples’ - computes the orbit of the list interpreted as an ordered tuple under the group action ( i.e., g((1,2,3)) = (g(1), g(2), g(3)) ) ‘sets’ - computes the orbit of the list interpreted as a sets

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 2, 0, 4, 5, 6, 3])
>>> G = PermutationGroup([a])
>>> G.orbit(0)
{0, 1, 2}
>>> G.orbit([0, 4], 'union')
{0, 1, 2, 3, 4, 5, 6}

See also

orbit_transversal

orbit_rep(alpha, beta, schreier_vector=None)[source]¶

Return a group element which sends alpha to beta.

Explanation

If beta is not in the orbit of alpha, the function returns False. This implementation makes use of the schreier vector. For a proof of correctness, see [1], p.80

Examples

>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> G = AlternatingGroup(5)
>>> G.orbit_rep(0, 4)
(0 4 1 2 3)

See also

schreier_vector

orbit_transversal(alpha, pairs=False)[source]¶

Computes a transversal for the orbit of alpha as a set.

Explanation

For a permutation group \(G\), a transversal for the orbit \(Orb = \{g(\alpha) | g \in G\}\) is a set \(\{g_\beta | g_\beta(\alpha) = \beta\}\) for \(\beta \in Orb\). Note that there may be more than one possible transversal. If pairs is set to True, it returns the list of pairs \((\beta, g_\beta)\). For a proof of correctness, see [1], p.79

Examples

>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> G = DihedralGroup(6)
>>> G.orbit_transversal(0)
[(5), (0 1 2 3 4 5), (0 5)(1 4)(2 3), (0 2 4)(1 3 5), (5)(0 4)(1 3), (0 3)(1 4)(2 5)]

See also

orbit

orbits(rep=False)[source]¶

Return the orbits of self, ordered according to lowest element in each orbit.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(1, 5)(2, 3)(4, 0, 6)
>>> b = Permutation(1, 5)(3, 4)(2, 6, 0)
>>> G = PermutationGroup([a, b])
>>> G.orbits()
[{0, 2, 3, 4, 6}, {1, 5}]

order()[source]¶

Return the order of the group: the number of permutations that can be generated from elements of the group.

The number of permutations comprising the group is given by len(group); the length of each permutation in the group is given by group.size.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup

>>> a = Permutation([1, 0, 2])
>>> G = PermutationGroup([a])
>>> G.degree
3
>>> len(G)
1
>>> G.order()
2
>>> list(G.generate())
[(2), (2)(0 1)]

>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.order()
6

See also

degree

pointwise_stabilizer(points, incremental=True)[source]¶

Return the pointwise stabilizer for a set of points.

Explanation

For a permutation group \(G\) and a set of points \(\{p_1, p_2,\ldots, p_k\}\), the pointwise stabilizer of \(p_1, p_2, \ldots, p_k\) is defined as \(G_{p_1,\ldots, p_k} = \{g\in G | g(p_i) = p_i \forall i\in\{1, 2,\ldots,k\}\}\) ([1],p20). It is a subgroup of \(G\).

Examples

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(7)
>>> Stab = S.pointwise_stabilizer([2, 3, 5])
>>> Stab.is_subgroup(S.stabilizer(2).stabilizer(3).stabilizer(5))
True

Notes

When incremental == True, rather than the obvious implementation using successive calls to .stabilizer(), this uses the incremental Schreier-Sims algorithm to obtain a base with starting segment - the given points.

polycyclic_group()[source]¶

Return the PolycyclicGroup instance with below parameters:

Explanation

pc_sequence : Polycyclic sequence is formed by collecting all the missing generators between the adjacent groups in the derived series of given permutation group.
pc_series : Polycyclic series is formed by adding all the missing generators of der[i+1] in der[i], where der represents the derived series.
relative_order : A list, computed by the ratio of adjacent groups in pc_series.

presentation(eliminate_gens=True)[source]¶

Return an \(FpGroup\) presentation of the group.

The algorithm is described in [1], Chapter 6.1.

random(af=False)[source]¶: Return a random group element

random_pr(gen_count=11, iterations=50, _random_prec=None)[source]¶

Return a random group element using product replacement.

Explanation

For the details of the product replacement algorithm, see _random_pr_init In random_pr the actual ‘product replacement’ is performed. Notice that if the attribute _random_gens is empty, it needs to be initialized by _random_pr_init.

See also

_random_pr_init

random_stab(alpha, schreier_vector=None, _random_prec=None)[source]¶

Random element from the stabilizer of alpha.

The schreier vector for alpha is an optional argument used for speeding up repeated calls. The algorithm is described in [1], p.81

See also

random_pr, orbit_rep

schreier_sims()[source]¶

Schreier-Sims algorithm.

Explanation

It computes the generators of the chain of stabilizers \(G > G_{b_1} > .. > G_{b1,..,b_r} > 1\) in which \(G_{b_1,..,b_i}\) stabilizes \(b_1,..,b_i\), and the corresponding s cosets. An element of the group can be written as the product \(h_1*..*h_s\).

We use the incremental Schreier-Sims algorithm.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.schreier_sims()
>>> G.basic_transversals
[{0: (2)(0 1), 1: (2), 2: (1 2)},
 {0: (2), 2: (0 2)}]

schreier_sims_incremental(base=None, gens=None, slp_dict=False)[source]¶

Extend a sequence of points and generating set to a base and strong generating set.

Parameters:

base

The sequence of points to be extended to a base. Optional parameter with default value [].

gens

The generating set to be extended to a strong generating set relative to the base obtained. Optional parameter with default value self.generators.

slp_dict

If \(True\), return a dictionary \({g: gens}\) for each strong generator \(g\) where \(gens\) is a list of strong generators coming before \(g\) in \(strong_gens\), such that the product of the elements of \(gens\) is equal to \(g\).

Returns:

(base, strong_gens)

base is the base obtained, and strong_gens is the strong generating set relative to it. The original parameters base, gens remain unchanged.

Examples

>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> A = AlternatingGroup(7)
>>> base = [2, 3]
>>> seq = [2, 3]
>>> base, strong_gens = A.schreier_sims_incremental(base=seq)
>>> _verify_bsgs(A, base, strong_gens)
True
>>> base[:2]
[2, 3]

Notes

This version of the Schreier-Sims algorithm runs in polynomial time. There are certain assumptions in the implementation - if the trivial group is provided, base and gens are returned immediately, as any sequence of points is a base for the trivial group. If the identity is present in the generators gens, it is removed as it is a redundant generator. The implementation is described in [1], pp. 90-93.

See also

schreier_sims, schreier_sims_random

schreier_sims_random(base=None, gens=None, consec_succ=10, _random_prec=None)[source]¶

Randomized Schreier-Sims algorithm.

Parameters:

base

The sequence to be extended to a base.

gens

The generating set to be extended to a strong generating set.

consec_succ

The parameter defining the probability of a wrong answer.

_random_prec

An internal parameter used for testing purposes.

Returns:

(base, strong_gens)

base is the base and strong_gens is the strong generating set relative to it.

Explanation

The randomized Schreier-Sims algorithm takes the sequence base and the generating set gens, and extends base to a base, and gens to a strong generating set relative to that base with probability of a wrong answer at most \(2^{-consec\_succ}\), provided the random generators are sufficiently random.

Examples

>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(5)
>>> base, strong_gens = S.schreier_sims_random(consec_succ=5)
>>> _verify_bsgs(S, base, strong_gens) 
True

Notes

The algorithm is described in detail in [1], pp. 97-98. It extends the orbits orbs and the permutation groups stabs to basic orbits and basic stabilizers for the base and strong generating set produced in the end. The idea of the extension process is to “sift” random group elements through the stabilizer chain and amend the stabilizers/orbits along the way when a sift is not successful. The helper function _strip is used to attempt to decompose a random group element according to the current state of the stabilizer chain and report whether the element was fully decomposed (successful sift) or not (unsuccessful sift). In the latter case, the level at which the sift failed is reported and used to amend stabs, base, gens and orbs accordingly. The halting condition is for consec_succ consecutive successful sifts to pass. This makes sure that the current base and gens form a BSGS with probability at least \(1 - 1/\text{consec\_succ}\).

See also

schreier_sims

schreier_vector(alpha)[source]¶

Computes the schreier vector for alpha.

Explanation

The Schreier vector efficiently stores information about the orbit of alpha. It can later be used to quickly obtain elements of the group that send alpha to a particular element in the orbit. Notice that the Schreier vector depends on the order in which the group generators are listed. For a definition, see [3]. Since list indices start from zero, we adopt the convention to use “None” instead of 0 to signify that an element does not belong to the orbit. For the algorithm and its correctness, see [2], pp.78-80.

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([2, 4, 6, 3, 1, 5, 0])
>>> b = Permutation([0, 1, 3, 5, 4, 6, 2])
>>> G = PermutationGroup([a, b])
>>> G.schreier_vector(0)
[-1, None, 0, 1, None, 1, 0]

See also

orbit

stabilizer(alpha)[source]¶

Return the stabilizer subgroup of alpha.

Explanation

The stabilizer of \(\alpha\) is the group \(G_\alpha = \{g \in G | g(\alpha) = \alpha\}\). For a proof of correctness, see [1], p.79.

Examples

>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> G = DihedralGroup(6)
>>> G.stabilizer(5)
PermutationGroup([
    (5)(0 4)(1 3)])

See also

orbit

property strong_gens¶

Return a strong generating set from the Schreier-Sims algorithm.

Explanation

A generating set \(S = \{g_1, g_2, \dots, g_t\}\) for a permutation group \(G\) is a strong generating set relative to the sequence of points (referred to as a “base”) \((b_1, b_2, \dots, b_k)\) if, for \(1 \leq i \leq k\) we have that the intersection of the pointwise stabilizer \(G^{(i+1)} := G_{b_1, b_2, \dots, b_i}\) with \(S\) generates the pointwise stabilizer \(G^{(i+1)}\). The concepts of a base and strong generating set and their applications are discussed in depth in [1], pp. 87-89 and [2], pp. 55-57.

Examples

>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(4)
>>> D.strong_gens
[(0 1 2 3), (0 3)(1 2), (1 3)]
>>> D.base
[0, 1]

strong_presentation()[source]¶

Return a strong finite presentation of group. The generators of the returned group are in the same order as the strong generators of group.

The algorithm is based on Sims’ Verify algorithm described in [1], Chapter 6.

Examples

>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> P = DihedralGroup(4)
>>> G = P.strong_presentation()
>>> P.order() == G.order()
True

See also

presentation, _verify

subgroup(gens)[source]¶: Return the subgroup generated by \(gens\) which is a list of elements of the group

subgroup_search(prop, base=None, strong_gens=None, tests=None, init_subgroup=None)[source]¶

Find the subgroup of all elements satisfying the property prop.

Parameters:

prop

The property to be used. Has to be callable on group elements and always return True or False. It is assumed that all group elements satisfying prop indeed form a subgroup.

base

A base for the supergroup.

strong_gens

A strong generating set for the supergroup.

tests

A list of callables of length equal to the length of base. These are used to rule out group elements by partial base images, so that tests[l](g) returns False if the element g is known not to satisfy prop base on where g sends the first l + 1 base points.

init_subgroup

if a subgroup of the sought group is known in advance, it can be passed to the function as this parameter.

Returns:

res

The subgroup of all elements satisfying prop. The generating set for this group is guaranteed to be a strong generating set relative to the base base.

Explanation

This is done by a depth-first search with respect to base images that uses several tests to prune the search tree.

Examples

>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup)
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> S = SymmetricGroup(7)
>>> prop_even = lambda x: x.is_even
>>> base, strong_gens = S.schreier_sims_incremental()
>>> G = S.subgroup_search(prop_even, base=base, strong_gens=strong_gens)
>>> G.is_subgroup(AlternatingGroup(7))
True
>>> _verify_bsgs(G, base, G.generators)
True

Notes

This function is extremely lengthy and complicated and will require some careful attention. The implementation is described in [1], pp. 114-117, and the comments for the code here follow the lines of the pseudocode in the book for clarity.

The complexity is exponential in general, since the search process by itself visits all members of the supergroup. However, there are a lot of tests which are used to prune the search tree, and users can define their own tests via the tests parameter, so in practice, and for some computations, it’s not terrible.

A crucial part in the procedure is the frequent base change performed (this is line 11 in the pseudocode) in order to obtain a new basic stabilizer. The book mentiones that this can be done by using .baseswap(...), however the current implementation uses a more straightforward way to find the next basic stabilizer - calling the function .stabilizer(...) on the previous basic stabilizer.

sylow_subgroup(p)[source]¶

Return a p-Sylow subgroup of the group.

The algorithm is described in [1], Chapter 4, Section 7

Examples

>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.named_groups import AlternatingGroup

>>> D = DihedralGroup(6)
>>> S = D.sylow_subgroup(2)
>>> S.order()
4
>>> G = SymmetricGroup(6)
>>> S = G.sylow_subgroup(5)
>>> S.order()
5

>>> G1 = AlternatingGroup(3)
>>> G2 = AlternatingGroup(5)
>>> G3 = AlternatingGroup(9)

>>> S1 = G1.sylow_subgroup(3)
>>> S2 = G2.sylow_subgroup(3)
>>> S3 = G3.sylow_subgroup(3)

>>> len1 = len(S1.lower_central_series())
>>> len2 = len(S2.lower_central_series())
>>> len3 = len(S3.lower_central_series())

>>> len1 == len2
True
>>> len1 < len3
True

property transitivity_degree¶

Compute the degree of transitivity of the group.

Explanation

A permutation group \(G\) acting on \(\Omega = \{0, 1, \dots, n-1\}\) is k-fold transitive, if, for any \(k\) points \((a_1, a_2, \dots, a_k) \in \Omega\) and any \(k\) points \((b_1, b_2, \dots, b_k) \in \Omega\) there exists \(g \in G\) such that \(g(a_1) = b_1, g(a_2) = b_2, \dots, g(a_k) = b_k\) The degree of transitivity of \(G\) is the maximum k such that \(G\) is k-fold transitive. ([8])

Examples

>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 2, 0])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.transitivity_degree
3

See also

is_transitive, orbit