# Definition A [[subgroup]] $N \subset G$, for some [[group]] $G$, is [[Conjugacy Class|self-conjugate]] (or normal, or invariant) if $g N g^{-1}$ is identical to $N$ for all possible choices of $g \in G$, i.e. $ g N g^{-1} \cong N, \quad \forall g \in G $ This is usually denoted $ N \lhd G $ To form a self-conjugate subgroup, one must include entire conjugacy classes in the subgroup. # Abelian Groups The subgroups of an [[Abelian group]] are self-conjugate subgroups. # Theorem The right and left [[Coset|cosets]] of a normal subgroup $N$ are the same. ### Proof $ gN = gNg^{-1} g = Ng \qquad \square $ where in the second equality we used the self-conjugacy of the matrix. # Theorem The multiplication of the elements of two right cosets of a normal subgroup gives another right coset. A similar theorem holds for left cosets. ^2f79f8 $ (Ng)(Nh) = (gN)(Nh) = (gN)(g^{-1} g) (N g^{-1}g h) = (N^2) (gh) = N (g h) $