Permutation Groups¶

class
sympy.combinatorics.perm_groups.
PermutationGroup
[source]¶ The class defining a Permutation group.
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.
References
[1] Holt, D., Eick, B., O’Brien, E. “Handbook of Computational Group Theory”
[2] Seress, A. “Permutation Group Algorithms”
[3] http://en.wikipedia.org/wiki/Schreier_vector
[4] http://en.wikipedia.org/wiki/Nielsen_transformation #Product_replacement_algorithm
[5] Frank Celler, Charles R.LeedhamGreen, Scott H.Murray, Alice C.Niemeyer, and E.A.O’Brien. “Generating Random Elements of a Finite Group”
[6] http://en.wikipedia.org/wiki/Block_%28permutation_group_theory%29
[7] http://www.algorithmist.com/index.php/Union_Find
[8] http://en.wikipedia.org/wiki/Multiply_transitive_group#Multiply_transitive_groups
[9] http://en.wikipedia.org/wiki/Center_%28group_theory%29
[10] http://en.wikipedia.org/wiki/Centralizer_and_normalizer
[11] http://groupprops.subwiki.org/wiki/Derived_subgroup
[12] http://en.wikipedia.org/wiki/Nilpotent_group
[13] http://www.math.colostate.edu/~hulpke/CGT/cgtnotes.pdf
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.permutations import Cycle >>> from sympy.combinatorics.polyhedron import Polyhedron >>> from sympy.combinatorics.perm_groups import PermutationGroup
The permutations corresponding to motion of the front, right and bottom face of a 2x2 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 2x2 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']

base
¶ Return a base from the SchreierSims algorithm.
For a permutation group \(G\), a base is a sequence of points \(B = (b_1, b_2, ..., b_k)\) 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. 8789 and [2], pp. 5557.
An alternative way to think of \(B\) is that it gives the indices of the stabilizer cosets that contain more than the identity permutation.
See also
strong_gens
,basic_transversals
,basic_orbits
,basic_stabilizers
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.
If a base for a group \(G\) is given by \((b_1, b_2, ..., b_k)\), this function returns a base \((b_1, b_2, ..., b_{i+1}, b_i, ..., b_k)\), 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.
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, andstrong_gens
is a generating set relative to it.See also
Notes
The deterministic version of the algorithm is discussed in [1], pp. 102103; 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.
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

basic_orbits
¶ Return the basic orbits relative to a base and strong generating set.
If \((b_1, b_2, ..., b_k)\) is a base for a group \(G\), and \(G^{(i)} = G_{b_1, b_2, ..., b_{i1}}\) is the
i
th basic stabilizer (so that \(G^{(1)} = G\)), thei
th basic orbit relative to this base is the orbit of \(b_i\) under \(G^{(i)}\). See [1], pp. 8789 for more information.See also
Examples
>>> from sympy.combinatorics.named_groups import SymmetricGroup >>> S = SymmetricGroup(4) >>> S.basic_orbits [[0, 1, 2, 3], [1, 2, 3], [2, 3]]

basic_stabilizers
¶ Return a chain of stabilizers relative to a base and strong generating set.
The
i
th basic stabilizer \(G^{(i)}\) relative to a base \((b_1, b_2, ..., b_k)\) is \(G_{b_1, b_2, ..., b_{i1}}\). For more information, see [1], pp. 8789.See also
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)])

basic_transversals
¶ Return basic transversals relative to a base and strong generating set.
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. 8789 for more information.
See also
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.
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]).
See also
Notes
This is a naive implementation that is a straightforward application of
.centralizer()
Examples
>>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.named_groups import DihedralGroup >>> D = DihedralGroup(4) >>> G = D.center() >>> G.order() 2

centralizer
(other)[source]¶ Return the centralizer of a group/set/element.
The centralizer of a set of permutations
S
inside a groupG
is the set of elements ofG
that commute with all elements ofS
:`C_G(S) = \{ g \in G  gs = sg \forall s \in S\}` ([10])
Usually,
S
is a subset ofG
, but ifG
is a proper subgroup of the full symmetric group, we allow forS
to have elements outsideG
.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.Parameters: other
a permutation group/list of permutations/single permutation
See also
Notes
The implementation is an application of
.subgroup_search()
with tests using a specific base for the groupG
.Examples
>>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... CyclicGroup) >>> S = SymmetricGroup(6) >>> C = CyclicGroup(6) >>> H = S.centralizer(C) >>> H.is_subgroup(C) True

commutator
(G, H)[source]¶ Return the commutator of two subgroups.
For a permutation group
K
and subgroupsG
,H
, the commutator ofG
andH
is defined as the group generated by all the commutators \([g, h] = hgh^{1}g^{1}\) forg
inG
andh
inH
. It is naturally a subgroup ofK
([1], p.27).See also
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)
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

contains
(g, strict=True)[source]¶ Test if permutation
g
belong to self,G
.If
g
is an element ofG
it can be written as a product of factors drawn from the cosets ofG
’s stabilizers. To see ifg
is one of the actual generators defining the group useG.has(g)
.If
strict
is notTrue
,g
will be resized, if necessary, to match the size of permutations inself
.See also
coset_factor
,has
,in
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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 ofg
If
g
is an element ofG
then it can be written as the product of permutations drawn from the SchreierSims coset decomposition,The permutations returned in
f
are those for which the product givesg
:g = f[n]*...f[1]*f[0]
wheren = len(B)
andB = G.base
. f[i] is one of the permutations inself._basic_orbits[i]
.If factor_index==True, returns a tuple
[b[0],..,b[n]]
, whereb[i]
belongs toself._basic_orbits[i]
Examples
>>> from sympy.combinatorics import Permutation, PermutationGroup >>> Permutation.print_cyclic = True >>> 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 util._strip

coset_rank
(g)[source]¶ rank using SchreierSims representation
The coset rank of
g
is the ordering number in which it appears in the lexicographic listing according to the coset decompositionThe ordering is the same as in G.generate(method=’coset’). If
g
does not belong to the group it returns None.See also
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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)

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

degree
¶ Returns the size of the permutations in the group.
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 bygroup.order()
.See also
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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)]

derived_series
()[source]¶ Return the derived series for the group.
The derived series for a group \(G\) is defined as \(G = G_0 > G_1 > G_2 > \ldots\) where \(G_i = [G_{i1}, G_{i1}]\), i.e. \(G_i\) is the derived subgroup of \(G_{i1}\), for \(i\in\mathbb{N}\). When we have \(G_k = G_{k1}\) for some \(k\in\mathbb{N}\), the series terminates.
Returns: A list of permutation groups containing the members of the derived
series in the order \(G = G_0, G_1, G_2, \ldots\).
See also
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

derived_subgroup
()[source]¶ Compute the derived subgroup.
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]).
See also
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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]]

elements
¶ Returns all the elements of the permutation group as a set
Examples
>>> from sympy.combinatorics import Permutation, PermutationGroup >>> p = PermutationGroup(Permutation(1, 3), Permutation(1, 2)) >>> p.elements {(3), (2 3), (3)(1 2), (1 2 3), (1 3 2), (1 3)}

generate
(method='coset', af=False)[source]¶ Return iterator to generate the elements of the group
Iteration is done with one of these methods:
method='coset' using the SchreierSims coset representation method='dimino' using the Dimino method
If af = True it yields the array form of the permutations
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> 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
References
[1] The Implementation of Various Algorithms for Permutation Groups in the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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]]

generate_schreier_sims
(af=False)[source]¶ Yield group elements using the SchreierSims representation in coset_rank order
If
af = True
it yields the array form of the permutationsExamples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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 \([s1, ..., sn]\) s.t \(g = sn*...*s1\). If \(original=True\), make the list contain only the original group generators

generators
¶ Returns the generators of the group.
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> a = Permutation([0, 2, 1]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G.generators [(1 2), (2)(0 1)]

identity
¶ Return the identity element of the permutation group.

is_abelian
¶ Test if the group is Abelian.
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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.
More specifically, it is onesided 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.
See also
_check_cycles_alt_sym
Notes
The algorithm itself uses some nontrivial results from group theory and number theory: 1) If a transitive group
G
of degreen
contains an element with a cycle of lengthn/2 < p < n2
forp
a prime,G
is the symmetric or alternating group ([1], pp. 8182) 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. 226227). 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.perm_groups import PermutationGroup >>> from sympy.combinatorics.named_groups import DihedralGroup >>> D = DihedralGroup(10) >>> D.is_alt_sym() False

is_nilpotent
¶ Test if the group is nilpotent.
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]).
See also
Examples
>>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... CyclicGroup) >>> C = CyclicGroup(6) >>> C.is_nilpotent True >>> S = SymmetricGroup(5) >>> S.is_nilpotent False

is_normal
(gr, strict=True)[source]¶ Test if
G=self
is a normal subgroup ofgr
.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 >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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

is_primitive
(randomized=True)[source]¶ Test if a group is primitive.
A permutation group
G
acting on a setS
is called primitive ifS
contains no nontrivial block under the action ofG
(a block is nontrivial if its cardinality is more than1
).See also
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 of0
. This algorithm has complexity \(O(n^2)\) wheren
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 \(G_0\) usingrandom_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 therandomized
flag.Examples
>>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.named_groups import DihedralGroup >>> D = DihedralGroup(10) >>> D.is_primitive() False

is_solvable
¶ Test if the group is solvable.
G
is solvable if its derived series terminates with the trivial group ([1], p.29).See also
Examples
>>> from sympy.combinatorics.named_groups import SymmetricGroup >>> S = SymmetricGroup(3) >>> S.is_solvable True

is_subgroup
(G, strict=True)[source]¶ Return
True
if all elements ofself
belong toG
.If
strict
isFalse
then ifself
’s degree is smaller thanG
’s, the elements will be resized to have the same degree.Examples
>>> from sympy.combinatorics import Permutation, PermutationGroup >>> from sympy.combinatorics.named_groups 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
toFalse
:>>> 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

is_transitive
(strict=True)[source]¶ Test if the group is transitive.
A group is transitive if it has a single orbit.
If
strict
isFalse
the group is transitive if it has a single orbit of length different from 1.Examples
>>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.perm_groups import 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

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 >>> from sympy.combinatorics.perm_groups import 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 \(G_k = [G, G_{k1}]\), 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 \(G = G_0, G_1, G_2, \ldots\) See also
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

make_perm
(n, seed=None)[source]¶ Multiply
n
randomly selected permutations from pgroup together, starting with the identity permutation. Ifn
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.See also
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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)

max_div
¶ Maximum proper divisor of the degree of a permutation group.
See also
minimal_block
,_union_find_merge
Notes
Obviously, this is the degree divided by its minimal proper divisor (larger than
1
, if one exists). As it is guaranteed to be prime, thesieve
fromsympy.ntheory
is used. This function is also used as an optimization tool for the functionsminimal_block
and_union_find_merge
.Examples
>>> from sympy.combinatorics import Permutation >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> G = PermutationGroup([Permutation([0, 2, 1, 3])]) >>> G.max_div 2

minimal_block
(points)[source]¶ For a transitive group, finds the block system generated by
points
.If a group
G
acts on a setS
, a nonempty subsetB
ofS
is called a block under the action ofG
if for allg
inG
we havegB = B
(g
fixesB
) orgB
andB
have no common points (g
movesB
entirely). ([1], p.23; [6]).The distinct translates
gB
of a blockB
forg
inG
partition the setS
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 dividesS
([1], p.23). AG
congruence is an equivalence relation~
on the setS
such thata ~ b
impliesg(a) ~ g(b)
for allg
inG
. For a transitive group, the equivalence classes of aG
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_{k1})
which is the same as finding the maximal block system (i.e., the one with minimum block size) such thatp_0, ..., p_{k1}
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 unionfind data structure. The running time is just above \(O(pointsS)\). ([1], pp. 8387; [7]).See also
_union_find_rep
,_union_find_merge
,is_transitive
,is_primitive
Examples
>>> from sympy.combinatorics.perm_groups import PermutationGroup >>> 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 \(False\).
See also
Examples
>>> from sympy.combinatorics import Permutation >>> from sympy.combinatorics.perm_groups import 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.
If
S
is a subset of a groupG
, the normal closure ofA
inG
is defined as the intersection of all normal subgroups ofG
that containA
([1], p.14). Alternatively, it is the group generated by the conjugatesx^{1}yx
forx
a generator ofG
andy
a generator of the subgroup\left\langle S\right\rangle
generated byS
(for some chosen generating set for\left\langle S\right\rangle
) ([1], p.73).Parameters: other
a subgroup/list of permutations/single permutation
k
an implementationspecific parameter that determines the number of conjugates that are adjoined to
other
at onceSee also
Notes
The algorithm is described in [1], pp. 7374; it makes use of the generation of random elements for permutation groups by the product replacement algorithm.
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

orbit
(alpha, action='tuples')[source]¶ Compute the orbit of alpha \(\{g(\alpha)  g \in G\}\) as a set.
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. 1921. 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
See also
Examples
>>> from sympy.combinatorics import Permutation >>> from sympy.combinatorics.perm_groups import 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}

orbit_rep
(alpha, beta, schreier_vector=None)[source]¶ Return a group element which sends
alpha
tobeta
.If
beta
is not in the orbit ofalpha
, the function returnsFalse
. This implementation makes use of the schreier vector. For a proof of correctness, see [1], p.80See also
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> G = AlternatingGroup(5) >>> G.orbit_rep(0, 4) (0 4 1 2 3)

orbit_transversal
(alpha, pairs=False)[source]¶ Computes a transversal for the orbit of
alpha
as a set.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 toTrue
, it returns the list of pairs \((\beta, g_\beta)\). For a proof of correctness, see [1], p.79See also
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> 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)]

orbits
(rep=False)[source]¶ Return the orbits of
self
, ordered according to lowest element in each orbit.Examples
>>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.perm_groups import 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 bygroup.size
.See also
Examples
>>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.perm_groups import 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

pointwise_stabilizer
(points, incremental=True)[source]¶ Return the pointwise stabilizer for a set of points.
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 \(G\).
See also
Notes
When incremental == True, rather than the obvious implementation using successive calls to
.stabilizer()
, this uses the incremental SchreierSims algorithm to obtain a base with starting segment  the given points.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

presentation
(eliminate_gens=True)[source]¶ Return an \(FpGroup\) presentation of the group.
The algorithm is described in [1], Chapter 6.1.

random_pr
(gen_count=11, iterations=50, _random_prec=None)[source]¶ Return a random group element using product replacement.
For the details of the product replacement algorithm, see
_random_pr_init
Inrandom_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

schreier_sims
()[source]¶ SchreierSims algorithm.
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 SchreierSims algorithm.
Examples
>>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.perm_groups import 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, andstrong_gens
is the strong generating set relative to it. The original parametersbase
,gens
remain unchanged.See also
Notes
This version of the SchreierSims algorithm runs in polynomial time. There are certain assumptions in the implementation  if the trivial group is provided,
base
andgens
are returned immediately, as any sequence of points is a base for the trivial group. If the identity is present in the generatorsgens
, it is removed as it is a redundant generator. The implementation is described in [1], pp. 9093.Examples
>>> from sympy.combinatorics.named_groups import AlternatingGroup >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> 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]

schreier_sims_random
(base=None, gens=None, consec_succ=10, _random_prec=None)[source]¶ Randomized SchreierSims algorithm.
The randomized SchreierSims algorithm takes the sequence
base
and the generating setgens
, and extendsbase
to a base, andgens
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.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 andstrong_gens
is the strong generating set relative to it.See also
Notes
The algorithm is described in detail in [1], pp. 9798. It extends the orbits
orbs
and the permutation groupsstabs
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 amendstabs
,base
,gens
andorbs
accordingly. The halting condition is forconsec_succ
consecutive successful sifts to pass. This makes sure that the currentbase
andgens
form a BSGS with probability at least \(1  1/\text{consec\_succ}\).Examples
>>> from sympy.combinatorics.perm_groups import PermutationGroup >>> 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) #doctest: +SKIP True

schreier_vector
(alpha)[source]¶ Computes the schreier vector for
alpha
.The Schreier vector efficiently stores information about the orbit of
alpha
. It can later be used to quickly obtain elements of the group that sendalpha
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 doesn’t belong to the orbit. For the algorithm and its correctness, see [2], pp.7880.See also
Examples
>>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.permutations import Permutation >>> 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]

stabilizer
(alpha)[source]¶ Return the stabilizer subgroup of
alpha
.The stabilizer of \(\alpha\) is the group \(G_\alpha = \{g \in G  g(\alpha) = \alpha\}\). For a proof of correctness, see [1], p.79.
See also
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.named_groups import DihedralGroup >>> G = DihedralGroup(6) >>> G.stabilizer(5) PermutationGroup([ (5)(0 4)(1 3)])

strong_gens
¶ Return a strong generating set from the SchreierSims algorithm.
A generating set \(S = \{g_1, g_2, ..., g_t\}\) for a permutation group \(G\) is a strong generating set relative to the sequence of points (referred to as a “base”) \((b_1, b_2, ..., b_k)\) if, for \(1 \leq i \leq k\) we have that the intersection of the pointwise stabilizer \(G^{(i+1)} := G_{b_1, b_2, ..., b_i}\) 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. 8789 and [2], pp. 5557.
See also
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 \(G\). The generators of the returned group are in the same order as the strong generators of \(G\).
The algorithm is based on Sims’ Verify algorithm described in [1], Chapter 6.
See also
presentation
,_verify
Example
>>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.named_groups import DihedralGroup >>> P = DihedralGroup(4) >>> G = P.strong_presentation() >>> P.order() == G.order() True

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
.This is done by a depthfirst search with respect to base images that uses several tests to prune the search tree.
Parameters: prop
The property to be used. Has to be callable on group elements and always return
True
orFalse
. It is assumed that all group elements satisfyingprop
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 thattests[l](g)
returns False if the elementg
is known not to satisfy prop base on where g sends the firstl + 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 basebase
.Notes
This function is extremely lengthy and complicated and will require some careful attention. The implementation is described in [1], pp. 114117, 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.Examples
>>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... AlternatingGroup) >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> 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

sylow_subgroup
(p)[source]¶ Return a pSylow 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

transitivity_degree
¶ Compute the degree of transitivity of the group.
A permutation group \(G\) acting on \(\Omega = \{0, 1, ..., n1\}\) is
k
fold transitive, if, for any k points \((a_1, a_2, ..., a_k)\in\Omega\) and any k points \((b_1, b_2, ..., b_k)\in\Omega\) there exists \(g\in G\) such that \(g(a_1)=b_1, g(a_2)=b_2, ..., g(a_k)=b_k\) The degree of transitivity of \(G\) is the maximumk
such that \(G\) isk
fold transitive. ([8])See also
Examples
>>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.permutations import Permutation >>> a = Permutation([1, 2, 0]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G.transitivity_degree 3
