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Let $G$ be a non-abelian group, let $\Pi$ be a group, and let $\eta: \Pi\rightarrow Out(G)$ be a homomorphism, where $Out(G)$ is the group of automorphisms of G modulo the normal subgroup of inner automorphisms. The obstruction to the existence of an exact sequence

$1\rightarrow G \rightarrow B \rightarrow \Pi \rightarrow 1$

that induces $\eta$ is a certain cohomology class in $H^3(\Pi,Z(G))$, where $Z(G)$ is the center of $G$ (c.f. Homology, Mac Lane, IV.8.).

Does anyone know an example of $(G,\Pi, \eta)$ where this obstruction is nonzero, with $G$ finite? What is the smallest group $G$ for which there exist $\Pi,\eta$ with a non-trivial obstruction?

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    $\begingroup$ I think the smallest example is when $G$ is the dihedral group of order 16. It has Out $G\cong C_2^2$, but there is no extension of $C_2^2$ by $G$ with this action. It must be in the literature somewhere, I suppose(?), so hopefully someone can give a reference. If not, I can post the computation as an answer $\endgroup$ Oct 25, 2013 at 22:10
  • $\begingroup$ @TimDokchitser, I'd be interested in this computation! $\endgroup$ Oct 27, 2013 at 23:50

1 Answer 1

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The smallest example when $\eta: \Pi\to \text{Out}(G)$ does not give rise to an extension of $\Pi$ by $G$ is when $G=D_{16}$, the dihedral group has order 16. (Apologies to those who write it as $D_8$.) Pick $c,h\in D_{16}$ a rotation of order 8 and a reflection. The elements of $\text{Out}(D_{16})\cong C_2\times C_2$ correspond to four types of automorphisms $$ \begin{array}{cccccccc} I &:&& c\mapsto c^{\pm 1} && h\mapsto h\cdot c^{\text{even}} && \text{(inner)}\cr II &:&& c\mapsto c^{\pm 3} && h\mapsto h\cdot c^{\text{even}} \cr III &:&& c\mapsto c^{\pm 1} && h\mapsto h\cdot c^{\text{odd}} \cr IV &:&& c\mapsto c^{\pm 3} && h\mapsto h\cdot c^{\text{odd}}, \cr \end{array} $$ and there is no extension of $\Pi=C_2$ by $G=D_{16}$ with $\Pi$ acting on $G$ as a type $IV$ automorphism.

One way to see this is just to go through the 51 groups of order 32; only five of them contain $D_{16}$, number 18,19,39,42,43 in GAP or Magma. The first two $D_{32}, SD_{32}$ give a type III action, next two $C_2\times D_{16}, (C_2\times C_8):C_2$ a type I action, and the last one $D_8:C_2^2$ a type II action.

Another way is to use the fact that dihedral, semi-dihedral and generalized quaternion groups are the only non-abelian 2-groups whose commutator subgroup has index 4 (and index 1 or 2 is impossible). The commutator subgroup $G'$ of $G=D_{16}$ is generated by $c^2$, and an extension $B$ of type $III$ or $IV$ acts non-trivially on $G/G'=\{1,c,h,ch\}$, so $B'<B$ has again index 4. So it must be $D_{32}$ or $SD_{32}$, since $Q_{32}$ has no dihedral subgroups. But both of these are visibly not type $IV$ extensions, because the non-trivial coset of $G$ in $B$ is represented by an element whose square is $c$, and it conjugates $c$ to itself.

I suppose in this example $$ H^3(\Pi,Z(G))=H^3(C_2,C_2)=H^1(C_2,C_2)=\text{Hom}(C_2,C_2)=C_2, $$ which pins down the obstruction class uniquely.

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  • $\begingroup$ This confused me for a second - I guess the idea is that the map $\Pi \to \mathrm{Out}(G)$ does not lift to a map $\Pi \to \mathrm{Aut}(G)$? $\endgroup$ Oct 30, 2013 at 13:22
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    $\begingroup$ No, not quite, this is subtle: even if there is no map $\Pi\to\text{Aut}G$, there might still be an extension. $\endgroup$ Oct 30, 2013 at 14:50

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