Permutation Groups¶

class diofant.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.

Examples

>>> Permutation.print_cyclic = True


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']


References

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

 Seress, A. “Permutation Group Algorithms”

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

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

__contains__(i)[source]

Return True if $$i$$ is contained in PermutationGroup.

Examples

>>> p = Permutation(1, 2, 3)
>>> Permutation(3) in PermutationGroup(p)
True

__eq__(other)[source]

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

Examples

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

__mul__(other)[source]

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

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

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

static __new__(cls, *args, **kwargs)[source]

The default constructor. Accepts Cycle and Permutation forms. Removes duplicates unless dups keyword is False.

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

Initialize random generators for the product replacement algorithm.

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

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 (). For practical purposes however, the values n = 50, r = 11 are suggested in .

Notes

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

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

Merges two classes in a union-find data structure.

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

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.

References

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

_union_find_rep(num, parents)[source]

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

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

Notes

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

References

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

property base

Return a base from the Schreier-Sims 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 , pp. 87-89 and , 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

>>> 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, and strong_gens is a generating set relative to it.

Examples

>>> from diofant.combinatorics.testutil import _verify_bsgs
>>> 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],
[Permutation(0, 1, 2, 3), Permutation(3)(0, 1), Permutation(1, 3, 2),
Permutation(2, 3), Permutation(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 , pp. 102-103; the randomized version is discussed in , p.103, and , p.98. It is of Las Vegas type. Notice that  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.

property 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_{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 , pp. 87-89 for more information.

Examples

>>> 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.

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

Examples

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

property 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 , pp. 87-89 for more information.

Examples

>>> A = AlternatingGroup(4)
>>> A.basic_transversals
[{0: Permutation(3),
1: Permutation(3)(0, 1, 2),
2: Permutation(3)(0, 2, 1),
3: Permutation(0, 3, 1)},
{1: Permutation(3),
2: Permutation(1, 2, 3),
3: Permutation(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 ().

Examples

>>> D = DihedralGroup(4)
>>> G = D.center()
>>> G.order()
2


Notes

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

centralizer(other)[source]

Return the centralizer of a group/set/element.

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\} ()


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.

Parameters

other – a permutation group/list of permutations/single permutation

Examples

>>> 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.

commutator(G, H)[source]

Return the commutator of two subgroups.

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 (, p.27).

Examples

>>> 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 (, p.28)

contains(g, strict=True)[source]

Test if permutation g belong to self, G.

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

>>> Permutation.print_cyclic = True

>>> a = Permutation(1, 2)
>>> b = Permutation(2, 3, 1)
>>> G = PermutationGroup(a, b, degree=5)
>>> G.contains(G)  # 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

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*f 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,..,b[n]], where b[i] belongs to self._basic_orbits[i]

Examples

>>> 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*f*f == g
True
>>> f1 = G.coset_factor(g, True); f1
[0, 4, 4]
>>> tr = G.basic_transversals
>>> f == tr[f1]
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 Schreier-Sims representation

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

>>> 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])
>>> c = Permutation(7)(2, 4)(3, 5)
>>> G.coset_rank(c)
16
>>> G.coset_unrank(16)
Permutation(7)(2, 4)(3, 5)

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.

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

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

Returns

• A list of permutation groups containing the members of the derived

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

Examples

>>> A = AlternatingGroup(5)
>>> len(A.derived_series())
1
>>> S = SymmetricGroup(4)
>>> len(S.derived_series())
4
>>> S.derived_series().is_subgroup(AlternatingGroup(4))
True
>>> S.derived_series().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 (, p.28, ).

Examples

>>> Permutation.print_cyclic = True
>>> 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]]

property elements

Returns all the elements of the permutation group in a list

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


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

Examples

>>> Permutation.print_cyclic = True


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, G)
>>> J = PermutationGroup(list(H.generate())); J
PermutationGroup([
Permutation(0, 1)(2, 3),
Permutation(3),
Permutation(1, 2, 3),
Permutation(1, 3, 2),
Permutation(0, 3, 1),
Permutation(0, 2, 3),
Permutation(0, 3)(1, 2),
Permutation(0, 1, 3),
Permutation(3)(0, 2, 1),
Permutation(0, 3, 2),
Permutation(3)(0, 1, 2),
Permutation(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

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

Examples

>>> Permutation.print_cyclic = True
>>> 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 Schreier-Sims representation in coset_rank order

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

Examples

>>> Permutation.print_cyclic = True
>>> 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]]

property generators

Returns the generators of the group.

Examples

>>> Permutation.print_cyclic = True
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.generators
[Permutation(1, 2), Permutation(2)(0, 1)]

property is_abelian

Test if the group is Abelian.

Examples

>>> Permutation.print_cyclic = True
>>> 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 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.

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 (, 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) (, p.82; , 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

>>> D = DihedralGroup(10)
>>> D.is_alt_sym()
False

property 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 (, p.29, ).

Examples

>>> 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 of gr.

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.generator and g2 g2 in G.generator

Examples

>>> Permutation.print_cyclic = True
>>> 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 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 , 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 , p.83). Behavior is changed by the randomized flag.

Examples

>>> D = DihedralGroup(10)
>>> D.is_primitive()
False

property is_solvable

Test if the group is solvable.

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

Examples

>>> S = SymmetricGroup(3)
>>> S.is_solvable
True

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

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))
False
>>> G3.is_subgroup(PermutationGroup(G3))
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

is_transitive(strict=True)[source]

Test if the group is transitive.

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

>>> 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

>>> 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 (, p.29).

Returns

• A list of permutation groups in the order

• G = G_0, G_1, G_2, \ldots

Examples

>>> A = AlternatingGroup(4)
>>> len(A.lower_central_series())
2
>>> A.lower_central_series().is_subgroup(DihedralGroup(2))
True

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

>>> Permutation.print_cyclic = True
>>> a, b = [Permutation([1, 0, 3, 2]), Permutation([1, 3, 0, 2])]
>>> G = PermutationGroup([a, b])
>>> G.make_perm(1, )
Permutation(0, 1)(2, 3)
>>> G.make_perm(3, [0, 1, 0])
Permutation(0, 2, 3, 1)
>>> G.make_perm([0, 1, 0])
Permutation(0, 2, 3, 1)

property max_div

Maximum proper divisor of the degree of a permutation group.

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, the sieve from diofant.ntheory is used. This function is also used as an optimization tool for the functions minimal_block and _union_find_merge.

Examples

>>> 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 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). (, p.23; ).

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| (, 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 (, 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 (, p.83).

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

Examples

>>> D = DihedralGroup(10)
>>> D.minimal_block([0, 5])
[0, 6, 2, 8, 4, 0, 6, 2, 8, 4]
>>> D.minimal_block([0, 1])
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

normal_closure(other, k=10)[source]

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

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 (, 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) (, p.73).

Parameters
• other – a subgroup/list of permutations/single permutation

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

Examples

>>> 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 , 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.

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 , p.78, , 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

>>> 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 to beta.

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 , p.80

Examples

>>> Permutation.print_cyclic = True
>>> G = AlternatingGroup(5)
>>> G.orbit_rep(0, 4)
Permutation(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 to True, it returns the list of pairs (\beta, g_\beta). For a proof of correctness, see , p.79

Examples

>>> Permutation.print_cyclic = True
>>> G = DihedralGroup(6)
>>> G.orbit_transversal(0)
[Permutation(5),
Permutation(0, 1, 2, 3, 4, 5),
Permutation(0, 5)(1, 4)(2, 3),
Permutation(0, 2, 4)(1, 3, 5),
Permutation(5)(0, 4)(1, 3),
Permutation(0, 3)(1, 4)(2, 5)]

orbits(rep=False)[source]

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

Examples

>>> 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

>>> a = Permutation([1, 0, 2])
>>> G = PermutationGroup([a])
>>> G.degree
3
>>> len(G)
1
>>> G.order()
2
>>> list(G.generate())
[Permutation(2), Permutation(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\}\} (,p20). It is a subgroup of G.

Examples

>>> 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.

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.

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.

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 , p.81

schreier_sims()[source]

Schreier-Sims 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 Schreier-Sims algorithm.

Examples

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

schreier_sims_incremental(base=None, gens=None)[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.

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 diofant.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 , pp. 90-93.

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

Randomized Schreier-Sims algorithm.

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.

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.

Examples

>>> from diofant.combinatorics.testutil import _verify_bsgs
>>> S = SymmetricGroup(5)
>>> base, strong_gens = S.schreier_sims_random(consec_succ=5)
>>> _verify_bsgs(S, base, strong_gens)  # doctest: +SKIP
True


Notes

The algorithm is described in detail in , 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}.

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 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 . 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 , pp.78-80.

Examples

>>> 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 , p.79.

Examples

>>> Permutation.print_cyclic = True
>>> G = DihedralGroup(6)
>>> G.stabilizer(5)
PermutationGroup([
Permutation(5)(0, 4)(1, 3),
Permutation(5)])

property strong_gens

Return a strong generating set from the Schreier-Sims 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 , pp. 87-89 and , pp. 55-57.

Examples

>>> D = DihedralGroup(4)
>>> D.strong_gens
[Permutation(0, 1, 2, 3), Permutation(0, 3)(1, 2), Permutation(1, 3)]
>>> D.base
[0, 1]


Find the subgroup of all elements satisfying the property prop.

This is done by a depth-first 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 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.

Examples

>>> from diofant.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 lenghty and complicated and will require some careful attention. The implementation is described in , 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.

property transitivity_degree

Compute the degree of transitivity of the group.

A permutation group G acting on \Omega = \{0, 1, ..., n-1\} 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 maximum k such that G is k-fold transitive. ()

Examples

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