Conjugate Subgroup

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, 2, ...) generates the so-called conjugate subgroup xHx^(-1). If, for all x, xHx^(-1)=H, then H is a normal (also called "self-conjugate" or "invariant") subgroup.

All subgroups of an Abelian group are normal.

See also

Normal Subgroup, Subgroup, Sylow Theorems

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Cite this as:

Weisstein, Eric W. "Conjugate Subgroup." From MathWorld--A Wolfram Web Resource.

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