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General Group Properties

General Group Properties

Subsections
IsAbelian(G) : GrpPC -> BoolElt
True if the group G is abelian, false otherwise.
IsConditioned(G) : GrpPC -> BoolElt
True if G has a conditioned presentation, false otherwise.
IsConsistent(G) : GrpPC -> BoolElt
True if G has a consistent presentation, false otherwise.
IsCyclic(G) : GrpPC -> BoolElt
True if the group G is cyclic, false otherwise.
IsElementaryAbelian(G) : GrpPC -> BoolElt
True if the group G is elementary abelian, false otherwise.
IsExtraSpecial(G) : GrpPC -> BoolElt
Given a p-group G, return true if G is extra-special, false otherwise.
IsNilpotent(G) : GrpPC -> BoolElt
True if the group G is nilpotent, false otherwise.
IsPerfect(G) : GrpPC -> BoolElt
True if the group G is perfect, false otherwise. A soluble group G is perfect only if it is trivial.
IsSimple(G) : GrpPC -> BoolElt
True if the group G is simple, false otherwise.
IsSoluble(G) : GrpPC -> BoolElt
IsSolvable(G) : GrpPC -> BoolElt
True if the group G is soluble, false otherwise. Thus, it always returns the value true for a pc-group.
IsSpecial(G) : GrpPC -> BoolElt
Given a p-group G, return true if G is special, false otherwise.

General Properties of Subgroups

IsCentral(G, H) : GrpPC, GrpPC -> BoolElt
True if the subgroup H of the group G lies in the centre of G, false otherwise.
IsConjugate(G, H, K) : GrpPC, GrpPC, GrpPC -> BoolElt, GrpPCElt
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 in the event that the subgroups are conjugate: an element z which conjugates H into K.
IsMaximal(G, H) : GrpPC, GrpPC -> BoolElt
True if the subgroup H of the group G is a maximal subgroup of G, false otherwise.
IsNormal(G, H) : GrpPC, GrpPC -> BoolElt
True if the subgroup H of the group G is a normal subgroup of G, false otherwise.
IsSelfNormalizing(G, H) : GrpPC, GrpPC -> BoolElt
True if the subgroup H of the group G is self-normalizing in G, false otherwise.
IsSubnormal(G, H) : GrpPC, GrpPC -> BoolElt
True if the subgroup H of the group G is subnormal in G, false otherwise.

Coercions Between Groups and Subgroups

G ! g : GrpPC, GrpPCElt -> GrpPCElt
Given an element g belonging to the subgroup H of the group G, rewrite g as an element of G.
H ! g : GrpPC, GrpPCElt -> GrpPCElt
Given an element g belonging to the group G, and given a subgroup H of G containing g, rewrite g as an element of H.
K ! g : GrpPC, GrpPCElt -> GrpPCElt
Given an element g belonging to the group H, and a group K, such that H and K are subgroups of G, and both H and K contain g, rewrite g as an element of K.
InclusionMap(G, H) : GrpPC, GrpPC -> Map
The map from the subgroup H of G to G.
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