## Normalizer and centralizer commute with inner automorphisms

Prove that if $G$ is a group, $S \subseteq G$, and $g \in G$, then $g N_G(S) g^{-1} = N_G(gSg^{-1})$ and $g C_G(S) g^{-1} = C_G(gSg^{-1})$.

Note that

• $x \in gN_G(S)g^{-1}$
• if and only if $g^{-1}xg \in N_G(S)$
• if and only if $(g^{-1}xg)S(g^{-1}xg) = S$
• if and only if $x(gSg^{-1})x^{-1} = gSg^{-1}$
• if and only if $x \in N_G(gSg^{-1})$,

and that

• $x \in g C_G(S) g^{-1}$
• if and only if $g^{-1}xg \in C_G(S)$
• if and only if $(g^{-1}xg)s(g^{-1}xg) = s$ for all $s \in S$
• if and only if $x(gsg^{-1})x^{-1} = gsg^{-1}$ for all $s \in S$
• if and only if $x \in C_G(gSg^{-1})$.