# What are central automorphisms used for?

A central automorphism is an automorphism $\theta$ for which $x^{-1}\theta(x)\in Z(G)$ for each $x\in G$.

It's not difficult to prove that the set of central automorphisms forms a subgroup of $\operatorname{Aut}(G)$ which I've seen written $\operatorname{Aut}_c(G)$.

Why are central automorphisms studied? What are they used for? Are they important conceptually, and if so, what is the intuition behind them? Is there anything special about how they work when $G$ is finite, nilpotent, or of prime order?

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Some basic facts. You can prove that $\theta \in \operatorname{Aut}(G)$ is a central automorphism if and only if $\theta$ commutes with every inner automorphism. Thus $\operatorname{Aut}_c(G)$ is the centralizer of $\operatorname{Inn}(G)$ in $\operatorname{Aut}(G)$. Also, there is a bijection between $\operatorname{Aut}_c(G)$ and $\operatorname{Hom}(G/G', Z(G))$. –  Mikko Korhonen Apr 13 '13 at 11:55

See, e.g.,

Bunina, E.I. Automorphisms of Chevalley groups of type $B_l$ over local rings with $1/2$. J. Math. Sci., New York 169, No. 5, 557-588 (2010); translation from Fundam. Prikl. Mat. 15, No. 7, 3-46 (2009).

Summary from Zentralblatt: We prove that every automorphism of a Chevalley group of type $B_l$, $l\ge  2$, over a commutative local ring with $1/2$ is standard, i.e., it is a composition of ring, inner, and central automorphisms.

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I think the intuition behind central automorphisms is that there are exactly those which commute with every inner automorphism of $G$.

A centerless finite group $G$ has no central automorphisms, so they are important only for groups with a non-trival center. Specifically they are important for studying finite nilpotent groups, particularly finite $p$-groups.

In general it is quite difficult to prove non-trivial results about the automorphism group of a finite $p$-group $G$, though the central automorphisms are relatively easier and can be used to investigate some properties of $\operatorname{Aut}(G)$. (You can find many examples by a simple search on the net.) This follows from the relation indicated above by m.k. : every central automorphism $\sigma$ determines a homomorphism $h_{\sigma}$ from $G$ to its center given by $h_{\sigma}(x) = x^{-1} \sigma (x)$. This induces an injective map $\sigma \rightarrow h_{\sigma}$, from $\operatorname{Aut}_c(G)$ to $\operatorname{Hom}(G,Z(G))$. This map would not be bijective in general (as indicated by m.k). A result of Adney and Yen asserts that it is bijective iff $G$ has no abelian direct factor (or $G$ is purely non-abelian).

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