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This didn't seem likely, so I looked for a counterexample and found one after a short search. Clearly $Q$ must not have any normal Sylow subgroups, so the first example to try is $Q=S_4$. Then $N$ mus …

$$\langle x,y \mid y^x=y^3, yx^{-1}y=x^{-5}\rangle.$$ I found that presentation by trial and error, but minimal presentations are known for $2$-groups of order up to $64$. See Sag, T. W.; Wamsley, J …

Let $S$ be a finite simple group and $V$ a faithful absolutely irreducible module for $S$. Then $W = V \otimes V$ is a faithful irreducible module for $S \times S$. Let $G = W \rtimes (S \times S)$ b …

A subgroup satisfying the condition $g \not\in H \Rightarrow H \cap g^{-1}Hg=1$ is called malnormal. This class of subgroups has been much studied. For many types of infinite groups, such as word-hyp …

In general, if $G=N \rtimes H$ is a semidirect product in which all nontrivial elements of $H$ act fixed-point-freely on $N$, then $H$ is malnormal in $G$. For example, if $K$ is any group and $H$ is …

I voted to close because I was unsure which way around the extension went but, as Yves said, the question is almost trivial if $C_p^n$ is the normal subgroup. So, suppose that $N \unlhd G$ with $N=C_ …

The answer is very similar to that of http://www.mathoverflow.net/questions/167446/ which was the case $e=1$. In fact it is a little more straightforward when $e>1$, because $p=2$ is no different from …

Let $H$ be extraspecial of order $p^3$ and exponent $p$ (for odd $p$), and $T=C_p$. Then $G=H \times T$ is capable but $T$ is not. We have $G = K/Z(K)$, where $$K= \langle a,b,c,d,e,f \mid [a,b]=c,[ …

The counterexample referred to in the comment by NAME_IN_CAPS is the split extension of the so-called deleted permutation module for $A_5$ over ${\mathbb F}_2$. that is, the $4$-dimensional irreducibl …

The proof is not difficult, and the argument can be found in older publications if you hunt around. I think it is essentially proved in a paper by Otto and Madlener. The point is that, if there is a …

Let $H = \times_{i=1}^\infty X_i$ be a direct product of countably many copies of a finite nonabelian group $X$, Take $G=H \rtimes C_2$, where $C_2 = \langle x \rangle$ acts on $H$ by permuting the co …

If I am understanding the question correctly, then I think the answer is no. (Did you mean "what about the category of locally finite group" rather than locally free groups?) Let $G$ and $H$ be defin …

$|G_2:G_3|$ can be an arbitrarily large finite number (although I don't know exactly what values it could take). If there is a finitely presented infinite simple group generated by elements of order $ …

Yes. In the paper P.J. Cameron, P.M. Neumann, and D.N. Teague, On the degrees of primitive permutation groups, Math. Z. 180 (1982), 141-149, the stronger statement is proved that, for almost all $n …

Added later: The example described below is not the smallest such example, which is $G={\rm PSU}(3,3)$ with $H \cong S_4$ and $\langle H,H^t \rangle \cong {\rm PSL}(2,7)$. But I will leave the origina …