The functions in this group provide access to basic information stored for a permutation group G.
The i-th defining generator for G. A negative subscript indicates that the inverse of the generator is to be created. The identity element of G will be created by G.0.
The degree of the permutation group G.
A set of elements of G that generate G.
The sequence of elements used to define the group G. Any occurrences of the identity element or any repetitions of a generator, as removed by Generators(G), are retained in this sequence. This function has the same effect as the expression [G.i : i in [1..Ngens(G)]] .
The number of defining generators for G.
A typically short sequence of random elements generating the group. Especially when groups are generated as subgroups, the result of FewGenerators is a much shorter sequence than returned by GeneratorsSequence.
The generic group containing G, i.e. the symmetric group in which G is naturally embedded.
The parent group G for the permutation g.
The natural G-set for the permutation group G.
> G := PermutationGroup< 12 | (1,6,7)(2,5,8,3,4,9)(11,12),
> (1,3)(4,9,12)(5,8,10,6,7,11) >;
> G;
Permutation group G acting on a set of cardinality 12
(1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12)
(1, 3)(4, 9, 12)(5, 8, 10, 6, 7, 11)
> G.1;
(1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12)
> G.1*G.2;
(1, 7, 3, 9, 2, 8)(4, 12, 5, 10, 6, 11)
> Degree(G);
12
> GSet(G);
GSet{ at 1 .. 12 atbrace
> Generic(G);
Symmetric group acting on a set of cardinality 12
Order = 479001600 = 2^10 * 3^5 * 5^2 * 7 * 11
> Generators(G);
{
(1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12),
(1, 3)(4, 9, 12)(5, 8, 10, 6, 7, 11)
}
> Ngens(G);
2
> x := G ! (1,6,7)(2,5,8,3,4,9)(11,12);
> x;
(1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12)
> Parent(x);
Permutation group G acting on a set of cardinality 12
Order = 648 = 2^3 * 3^4
(1, 6, 7)(2, 5, 8, 3, 4, 9)(11, 12)
(1, 3)(4, 9, 12)(5, 8, 10, 6, 7, 11)
]
Unless the order is already known, each of the functions in this family will create a base and strong generating set for the group if one does not already exist.
The order of the group G as an integer. If the order is not currently known, a base and strong generating set will be constructed for G.
The order of the group G returned as a factored integer. The factorization is returned in the form of a sequence Q which is defined as follows: If # G = p1e1 ... pnen, ei != 0, then Q will be the integer sequence [ <p1, e1>, ..., <pn, en> ]. If the order of G is not known, it will be computed.
Returns true if the group G is abelian, false otherwise.
Returns true if the group G is cyclic, false otherwise.
Returns true if the group G is elementary abelian, false otherwise.
Given a p-group G, return true if G is special, false otherwise.
Given a group G is a p-group G, return true if G is extra-special, false otherwise.
Returns true if the group G is nilpotent, false otherwise.
Returns true if the group G is soluble, false otherwise. Uses the algorithm of Sims [Sim90].
Returns true if the group G is perfect, false otherwise.
Returns true if the group G is simple, false otherwise.
Returns true if the group G is isomorphic to a wreath product A wreath B, where B is transitive, and false otherwise. If true, then three subgroups of G, call them A, B, C, are also returned. In this case we have G isomorphic to WreathProduct(A, CosetImage(B, C)).
> load m24; Loading "/home/magma/libs/pergps/m24" M24 - Mathieu group on 24 letters - degree 24 Order 244 823 040 = 2^10 * 3^3 * 5 * 7 * 11 * 23; Base 1,2,3,4,5,6,7 Group: G > time S := PerfectSubgroups(G); Time: 29.460 > [ Order(H) : R in S | not IsSimple(H) where H := R`subgroup ]; [ 120, 120, 120, 180, 180, 240, 240, 336, 336, 336, 336, 504, 720, 1008, 1080, 960, 960, 960, 1344, 1344, 1344, 1920, 2688, 2688, 2688, 2688, 2688, 2880, 3840, 3840, 5760, 10752, 11520, 11520, 40320, 21504, 21504, 32256, 64512, 69120, 322560 ]