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If you would like a complete classification following Jack's strategy, say your group is $G=C_n\rtimes C_m$ with $n,m$ coprime, $C_n=\langle c\rangle$, $C_m=\langle x\rangle$ and $xcx^{-1}=c^j$. The o …

I think a counterexample is $G=S_3\times S_3\times S_3$. Say $U$ is the "diagonal" $S_3$ in $G$; so if $G\lt S_9$ is generated by $(123)$, $(12)$, $(456)$, $(45)$ ,$(789)$ and $(78)$, then $U$ is g …

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 $D_8$ …

It's a great question! Disappointingly, I think the answer to (2) is No : The only restriction on a `good' division into "symmetric" vs. "symplectic" conjugacy classes that I can see is that it shoul …

Question: Given a finite group $G$, how do I find the smallest $n$ for which $G$ embeds in $S_n$? Equivalently, what is the smallest set $X$ on which $G$ acts faithfully by permutations? This lo …

Yes, this could happen: Say $p=2$, $l=3$, and take an elliptic curve $E/{\mathbb Q}_2$ with largest possible inertia image, $I=\text{SL}(2,{\mathbb F}_3)$. Then the $3$-adic representation $V_3(E)$ …

Yes, because $G$ and $Z/2Z$ have coprime order, every extension of one by the other splits by the Schur-Zassenhaus theorem (http://en.wikipedia.org/wiki/Schur-Zassenhaus_theorem). If you prefer, $H^2( …

For the first question, at least for $p=5, n=2$ there is a counterexample.: The group $\text{SL}_2({\mathbb F}_5)$ has a 2-dimensional symplectic representation with character in ${\mathbb Q}(\sqrt 5 …

Question: If you have a finite group, how do you name it? If, for whatever reason, you have to list all subgroups of $GL_2({\mathbb F}_5)$ up to isomorphism in a paper, you are likely to write so …