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?

  • 6
    $\begingroup$ 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 $\operatorname{Hom}(G/G', Z(G))$. $\endgroup$ Apr 13, 2013 at 11:55
  • $\begingroup$ @MikkoKorhonen what is $G^\prime$? $\endgroup$
    – Arrow
    Nov 17, 2016 at 13:49
  • $\begingroup$ @Arrow: The commutator subgroup of $G$. $\endgroup$ Nov 17, 2016 at 14:27

2 Answers 2


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).


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.