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 x h_i x^(-1), (i = 1, 2, ...) generates the so-called conjugate subgroup x H x^(-1). If, for all x, x H x^(-1) = H, then H is a normal (also called "self-conjugate" or "invariant") subgroup. All subgroups of an Abelian group are normal.

normal subgroup | subgroup | Sylow theorems