Subsections

• Abstract Group Predicates

Conjugate(H, g) : GrpFin, GrpFinElt -> GrpFin

Construct the conjugate g^(-1)Hg of the group H by the element g. The group H and the element g must belong to the same generic group.

Given groups H and K which belong to the same symmetric group, construct the intersection of H and K.

CommutatorSubgroup(H, K) : GrpFin, GrpFin -> GrpFin

Given groups H and K, both subgroups of the group G, construct the commutator subgroup of H and K in the group G. If K is a subgroup of H, then the group G may be omitted.

Centraliser(G, g) : GrpFin, GrpFinElt -> GrpFin

Construct the centralizer of the element g in the group G.

Centralizer(G, H) : GrpFin, GrpFin -> GrpFin

Centraliser(G, H) : GrpFin, GrpFin -> GrpFin

Construct the centralizer of the group H in the group G.

Given a subgroup H of the group G, construct the maximal normal subgroup of G that is contained in the subgroup H.

NormalClosure(G, H) : GrpFin, GrpFin -> GrpFin

Given a subgroup H of the group G, construct the normal closure of H in G.

Normaliser(G, H) : GrpFin, GrpFin -> GrpFin

Given a subgroup H of the group G, construct the normalizer of H in G.

Given a group G and a prime p dividing the order of G, construct the maximal normal p-subgroup of G.

Sylow(G, p) : GrpFin, RngIntElt -> GrpFin

Given a group G and a prime p, construct a Sylow p-subgroup of G.

True if the group G is abelian, false otherwise.

True if the group G is cyclic, false otherwise.

True if the group G is elementary abelian, false otherwise.

True if the subgroup H of the group G lies in the centre of G, false otherwise.

Given a group G and elements g and h belonging to G, return the value true if g and h are conjugate in G. The function returns a second value if the elements are conjugate: an element k which conjugates g into h.

IsConjugate(G, H, K) : GrpFin, GrpFin, GrpFin -> BoolElt, GrpFinElt

Given a group G and subgroups H and K belonging to G, return the value true if G and H are conjugate in G. The function returns a second value if the subgroups are conjugate: an element z which conjugates H into K.

Given a group G is a p-group G, return true if G is extra-special, false otherwise.

True if the subgroup H of the group G is a maximal subgroup of G. This function is evaluated by constructing the permutation representation of G on the cosets of H and testing this representation for primitivity. For this reason, the use of IsMaximal should be avoided if the index of H in G exceeds a one hundred thousand.

True if the group G is nilpotent, false otherwise.

True if the subgroup H of the group G is a normal subgroup of G, false otherwise.

True if the group G is perfect, false otherwise.

IsSelfNormalising(G, H) : GrpFin, GrpFin -> BoolElt

True if the subgroup H of the group G is self-normalizing in G, false otherwise.

True if the group G is simple, false otherwise.

IsSolvable(G) : GrpFin -> BoolElt

True if the group G is soluble, false otherwise.

Given a p-group G, return true if G is special, false otherwise.

True if the subgroup H of the group G is subnormal in G, false otherwise.

True if G is trivial, false otherwise.