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I'm looking for examples of outer automorphisms of a finite group $G$ which do not lift to automorphisms (i.e. non-split quotient map $\mathrm{Aut}(G)\to \mathrm{Out}(G)$, where $\mathrm{Out}(G) = \mathrm{Aut}(G)/\mathrm{Inn}(G)).$

I'm aware of the well-known $A_6$ (also $S_6$?) example, as explained in this Wikipedia article. The article also mentions $PSL(2.q^2)$ for $q$ odd.

Are these the smallest examples?

Is there a complete (or at least longer) list of examples somewhere?

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    $\begingroup$ I think that it is usually the case ( ie apart from small exceptions) that when $E$ is an extraspecial group of order $2^{2n+1}$ and $G = E *C$ (central product, with $C$ cyclic of order $4$), then ${\rm Out}(G) \cong {\rm Sp}(2n,2)$ but ${\rm Aut }(G)$ is not a split extension of the form you want. $\endgroup$
    – Geoff Robinson
    Commented Feb 20, 2018 at 15:30
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    $\begingroup$ For $\mathbb S_6$ it splits. $\endgroup$
    – Jarek Kuben
    Commented Feb 20, 2018 at 15:36
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    $\begingroup$ I think the $S_6$ remark was in reply to the parenthetical question in my second paragraph. Thanks for the answer. $\endgroup$
    – Kevin Walker
    Commented Feb 20, 2018 at 17:33
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    $\begingroup$ (And my previous comment was in response to a now deleted comment.) $\endgroup$
    – Kevin Walker
    Commented Feb 20, 2018 at 19:38
  • $\begingroup$ It's hard to choose but you have to choose one answer to accept. People helped you with votes, so it can help you. $\endgroup$
    – YCor
    Commented Feb 24, 2018 at 19:35

4 Answers 4

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As far I can remember, the smallest example is the dihedral group $G=D_{10}$ of order $10$.

In this case, $\mathrm{Aut}(G)$ is the Frobenius group of order $20$ and, since $Z(G)$ is trivial, $\mathrm{Inn}(G) \simeq G$, so $\mathrm{Out}(G)=\mathbb{Z}_2$.

It is easy to check that all elements of order $2$ in $\mathrm{Aut}(G)$ are actually contained in $\mathrm{Inn}(G)$, so there is no section $\mathrm{Out}(G) \to \mathrm{Aut}(G)$ and the automorphism sequence is non-split.

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  • $\begingroup$ Thanks for the helpful answer. Hard to choose just one answer to accept. $\endgroup$
    – Kevin Walker
    Commented Feb 20, 2018 at 18:23
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There is a complete description in

A. Lucchini, F. Menegazzo, and M. Morigi, On the existence of a complement for a finite simple group in its automorphism group, Illinois J. Math. Volume 47, Number 1-2 (2003), 395-418. (Project Euclid)

of which finite simple groups have complements in their automorphism groups.

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  • $\begingroup$ Thanks for the helpful answer. Hard to choose just one answer to accept. $\endgroup$
    – Kevin Walker
    Commented Feb 20, 2018 at 18:24
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As Francesco says, the smallest example is $D_{10}$. It is a normal subgroup of the Frobenius group of order $20$ and that extension is not split, as can be seen from looking at the $2$-Sylows. There are many more small examples, here is Magma code that will produce them for you:

for X in SmallGroups([2..100]) do
auts:=AutomorphismGroup(X);
G:=PermutationGroup(auts);
m:=PermutationRepresentation(auts);
H:=sub<G|[a: a in G | IsInner(a@@m)]>;
if not HasComplement(G,H) then GroupName(X), GroupName(G); end if;
end for;

The first few lines of the output:

D10 G20
D16 D8:C2^2
Q16 D8:C2^2
C2*D8 C2wrC2^2
C2*Q8 C2^3:S4
C3:S3 H432
C5:C4 C2*G20
D20 C2*G20
D26 C13:C12
C3*D10 C2*G20
C8:C4 (C2^2*C4):D8
C8:C4 C2^2:(D8:C2^2)
C8:C4 C2^2:(D8:C2^2)
H32^15 D8:C2^3
OMC32 D8:C2^2
D32 D16:(C2*C4)
Q32 D16:(C2*C4)

Instead of a (mostly) human-readable name, you can also get a unique identifier for the groups by replacing "GroupName" by "IndentifyGroup". That will allow you to play more with these examples, by allowing you to reconstruct them later from their identifiers.

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    $\begingroup$ There is a command HasComplement(G,H), which you could use to simplify that test. Also your computation of H can be very slow if G is large. $\endgroup$
    – Derek Holt
    Commented Feb 20, 2018 at 16:11
  • $\begingroup$ Thanks for the helpful answer. Hard to choose just one answer to accept. $\endgroup$
    – Kevin Walker
    Commented Feb 20, 2018 at 18:24
  • $\begingroup$ Thanks, Derek, I could not remember what it was called. $\endgroup$
    – Alex B.
    Commented Feb 20, 2018 at 18:43
  • $\begingroup$ @FrancescoPolizzi's answer referenced in this answer. $\endgroup$
    – LSpice
    Commented Aug 3, 2022 at 18:07
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A variation on Geoff's comment: Griess has proved that it's usually nonsplit for extraspecial $2$-groups:

Robert L. Griess, Jr., Automorphisms of extra special groups and nonvanishing degree 2 cohomology, Pacific Journal of Mathematics Vol. 48, No. 2 (1973) pp 403-422 doi:10.2140/pjm.1973.48.403, (pdf)

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  • $\begingroup$ no worries, happy to help. $\endgroup$
    – David Roberts ♦
    Commented Feb 23, 2018 at 22:01
  • $\begingroup$ @GeoffRobinson's comment referenced in this answer. $\endgroup$
    – LSpice
    Commented Aug 3, 2022 at 18:07

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