The Subgroup Structure

Although, in the case of an abelian group, many of the standard subgroup constructors are trivial, they are all implemented for the sake of uniformity.

H ^ g : GrpAb, GrpAbElt -> GrpAb

Conjugate(H, g) : GrpAb, GrpAbElt -> GrpAb

Construct the conjugate g^(-1) H g of the group H under the action of the element g. The group H and the element g must belong to a common group.

H ^ G : GrpAb, GrpAb -> GrpAb

ncl< G | H > : GrpAb, GrpAb -> GrpAb

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

H meet K : GrpAb, GrpAb -> GrpAb

Given subgroups H and K of some group G, construct their intersection.

H meet:= K : GrpAb, GrpAb -> GrpAb

Replace H with the intersection of groups H and K.

CommutatorSubgroup(G, H, K) : GrpAb, GrpAb, GrpAb -> GrpAb

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

Construct the commutator subgroup of groups H and K, where H and K are subgroups of a common group G.

Centralizer(G, g) : GrpAb, GrpAbElt -> GrpAb

Centraliser(G, g) : GrpAb, GrpAbElt -> GrpAb

The centralizer of the element g in the group G.

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

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

The centralizer of the group H in the group G.

Core(G, H) : GrpAb, GrpAb -> GrpAb

The maximal normal subgroup of G that is contained in the subgroup H of G.

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

The normal closure of the subgroup H in the group G.

Normalizer(G, H) : GrpAb, GrpAb -> GrpAb

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

The normalizer of the subgroup H of the group G.

SylowSubgroup(G, p) : GrpAb, RngIntElt -> GrpAb

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

The Sylow p-subgroup for the group G.

pCore(G, p) : GrpAb, RngIntElt -> GrpAb

The largest normal p-subgroup of the group G.