I'm looking to represent the automorphism group of a finite group in a small group theory package I'm writing.

I was thinking of representing a generic automorphism $a \in \operatorname{Aut}(G)$ as a pair $(o,i)$ representing the composition of an outer automorphism $o \in \operatorname{Out}(G)$ by an inner automorphism $i \in \operatorname{Inn}(G)$. Thus I can write $i$ as an element of $G / \operatorname{Center}(G)$, and use a set of coset representatives for $\operatorname{Out}(G)$. With this information, I can at least enumerate the elements of $\operatorname{Aut}(G)$.

Now, I'd like eventually to use the group structure on $\operatorname{Aut}(G)$. I guess that I'd need to write $\operatorname{Aut}(G)$ as a semidirect product to be able to compute the composition of elements of $\operatorname{Aut}(G)$ as represented by those pairs. Now, as some automorphism groups do not split in that way, I guess I'm out of luck?

I'm not so familiar with the group extension problem.


1 Answer 1


Yes, you are indeed out of luck when that extension is nonsplit! Representing ${\rm Aut}(G)$ for finite groups $G$ is one of the more difficult problems in computational group theory.

The default approach is to look for as small a subset of $G$ as you can find (unions of conjugacy classes are a good thing to try) on which ${\rm Aut}(G)$ acts faithfully, and to use that to get a permutation representation of ${\rm Aut}(G)$, but in difficult examples that can lead to very large degrees.

For finite simple groups $G$, it is useful to store suitable representation of ${\rm Aut}(G)$ - so for example, for $A_n$ with $n \ne 6$ you can use $S_n$.

Also, lots of finite solvable groups have solvable automorphism groups, and you can represent those using power-conjugate representations.


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.