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The Subgroup Structure

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.
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