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A subgroup is a subset H of group elements of a group G that satisfies the four group requirements. It must therefore contain the identity element. "H is a subgroup of G" is ...

A quantity which remains unchanged under certain classes of transformations. Invariants are extremely useful for classifying mathematical objects because they usually reflect ...

A subgroup H of an original group G has elements h_i. Let x be a fixed element of the original group G which is not a member of H. Then the transformation xh_ix^(-1), (i=1, ...

Let H be a subgroup of a group G. The similarity transformation of H by a fixed element x in G not in H always gives a subgroup. If xHx^(-1)=H for every element x in G, then ...

A type of maximal Abelian subgroup.

The fitting subgroup is the subgroup generated by all normal nilpotent subgroups of a group H, denoted F(H). In the case of a finite group, the subgroup generated will itself ...

A normalizer of a nontrivial Sylow p-subgroup of a group G.

An invariant series of a group G is a normal series I=A_0<|A_1<|...<|A_r=G such that each A_i<|G, where H<|G means that H is a normal subgroup of G.

The commutator subgroup (also called a derived group) of a group G is the subgroup generated by the commutators of its elements, and is commonly denoted G^' or [G,G]. It is ...

L is a subnormal subgroup of H if there is a "normal series" (in the sense of Jordan-Hölder) from L to H.