# What does it mean for an automorphism to centralize factor group $G/M$?

Let $G$ be a group and $M$ be a normal subgroup of it. An automorphism $\phi$ centralizes the factor group $G/M$.

What does it mean for an automorphism to centralize a factor group $G/M$?

I encountered the term here [http://arxiv.org/pdf/0803.4081]. In second paragraph he denote it by $\operatorname{Aut}^N (G)$.

• I've not seen the term before, but it seems relatively easy to guess that the condition means that $gM = \phi(g)M$ for all $g\in G$, and thus $\phi$ induces the identity automorphism of $G/M$ in the naive way. Of course this doesn't hold for arbitrary $\phi,M$. Where are you encountering this term? The intro to this paper seems to confirm my guess. – zibadawa timmy May 29 '16 at 23:15

Given $\phi\in\operatorname{Aut}(G)$ we say that $\phi$ centralizes a subgroup $H\subseteq G$ if $\phi(h)=h$ for all $h\in H$. So $\phi$ fixes $H$ elementwise. Equivalently, the restriction map gives a group homomorphism $\operatorname{Aut}(G)\to\operatorname{Aut}(H)$, and the automorphisms that centralize $H$ are exactly the kernel of this homomomorphism.
On the other hand, we say that $\phi\in\operatorname{Aut}(G)$ centralizes the quotient $G/M$ if and only if $gM=\phi(g)M$ for all $g\in G$. Equivalently, $\phi(g)g^{-1}\in M$ for all $g\in G$. Note, however, that the map $gM\mapsto \phi(g)M$ need not be a group homomorphism for arbitrary $\phi,M$, for this happens if and only if $\phi(M)\subseteq M.$ In particular, this map gives a group homomorphism for all $\phi\in\operatorname{Aut}(G)$ if and only if $M$ is characteristic in $G$ (by definition). However, those $\phi\in\operatorname{Aut}(G)$ with $\phi(M)\subseteq M$ form a subgroup $T_M$ of $\operatorname{Aut}(G)$, and we have a well-defined group homomorphism $T_M\to \operatorname{Aut}(G/M)$ given by $\phi\mapsto( gM\mapsto \phi(g)M)$, and the automorphisms of $G$ that centralize $G/M$ are precisely the kernel of this homomorphism. As noted before $T_M=\operatorname{Aut}(G)$ if and only if $M$ is characteristic in $G.$
For example, in the notation of the OP's reference $$\operatorname{Aut}^{Z(G)}(G) = \{ \phi\in\operatorname{Aut}(G) \ | \ \phi(g)g^{-1}\in Z(G) \ \forall g\in G\}$$ is the central automorphism group, which is precisely the centralizer subgroup $C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G))$, and $$\operatorname{Aut}_{Z(G)}(G) = \{ \phi\in\operatorname{Aut}(G) \ | \ \phi(z)=z \ \forall z\in Z(G) \}$$ are the automorphisms of $G$ that fix $Z(G)$ elementwise, and finally $$\operatorname{Aut}_{Z(G)}^{Z(G)}(G) = \operatorname{Aut}_{Z(G)}(G) \cap \operatorname{Aut}^{Z(G)}(G)$$ are the central automorphisms that fix the center elementwise.