Subsections

• General Properties of Subgroups
• Coercions Between Groups and Subgroups

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 group G is nilpotent, false otherwise.

True if the group G is perfect, false otherwise. A soluble group G is perfect only if it is trivial.

True if the group G is simple, false otherwise.

IsSolvable(G) : GrpAb -> BoolElt

True if the group G is soluble, false otherwise.

IsCentral(G, H) : GrpAb, GrpAb -> BoolElt

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

Given a group G and subgroups H and K belonging to G, return the value true if H and K 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.

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

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

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

Given an element g belonging to the subgroup H of the group G, rewrite g as an element of G.

H ! g : GrpAb, GrpAbElt -> GrpAbElt

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 : GrpAb, GrpAbElt -> GrpAbElt

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.

The inclusion monomorphism from the subgroup H of G into G.