2
$\begingroup$

A stem extension of a group $X$ is a group $G$ with a subgroup $N$ contained in $G' \cap Z(G)$ such that $G/N$ is isomorphic to $X$, so that we have a short exact sequence $1 \to N \to G \to X \to 1$.

A double cover of $X$ is a stem extension $1 \to N \to G \to X \to 1$ such that $|N|=2$.

My question is: if $1 \to N \to G \to X \to 1$ is any double cover of $X$ is it always the case that the (unique) generator of the cyclic group $N$ is a square in $G$? That is, writing $N=\langle z \rangle = \{1,z\}$ is there some $x \in G$ with $x^2 = z$?

I'm talking about the general case ($X$ not necessarily perfect). It would be nice to at least get, if not an answer, a feeling of how reasonable this question is. If you have a reference about the general concept of double cover defined above and its properties it would be greatly appreciated.

Thanks!

$\endgroup$
0

1 Answer 1

4
$\begingroup$

In SmallGroup(16,3), the derived subgroup has order 2, is central, and its generator is not a square. (See groupprops.subwiki.org/wiki/SmallGroup(16,3)#Subgroups)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.