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 $x h_i x^{-1}$, ($i=1$, 2, ...) generates a conjugate Subgroup $xHx^{-1}$. If, for all $x$, $xHx^{-1} = H$, then $H$ is a Self-Conjugate (also called Invariant or Normal) Subgroup. All Subgroups of an Abelian Group are invariant.