0
$\begingroup$

Let $ M $ be a finite group of order $2^{a+1} $ and let $ M $ have a normal subgroup $ R $ such that $|M: R|=2 $. Also we know that $ R $ is an elementary abelian subgroup of $ M $.

For example $ D_8 $ is an example for it. Is it true that for other cases $ M $ is abelian or what information is available for $ M $?

Thanks for your help

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ You can construct examples by taking central products of copies of $D_8$; you can also construct examples ex nihilo by starting with an $a$-dimensional vector space $V$ over $\mathbb{F}_2$, and an automorphism $\tau$ of order $2$ with at least one fixed point $v$ other than the identity. Then let $x$ act on $V$ as $\tau$, and set $x^2=v$; the resulting nonsplit extension of $V$ by $\mathbf{C}_2$ should have the properties you want. $\endgroup$
    – Arturo Magidin
    Commented Dec 1, 2013 at 21:10

1 Answer 1

Reset to default
3
$\begingroup$

The elementary Abelian $2$-group $R$ is a ${\rm GF}(2)M/R$-module of dimension $a$. As such it is a direct sum of $t$ copies of the trivial module and $\frac{a-t}{2}$-copies of the free module of rank $1$. Then $Z(M)$ has order $2^{t},$ and there are examples of $M$ and $R$ for which every possibility of $t$ between $1$ and $a$ occur, as Arturo comments.

Improve this answer
$\endgroup$

Not the answer you're looking for? Browse other questions tagged .