Definition A $2$-group $S$ is called wreathed if it is isomorphic to $(C\times C)\rtimes \langle i \rangle$ where $C$ is a cyclic group of order $2^n$ and $i$ is an involution with action $(a,b)^i=(b,a)$ for all $(a,b)\in C\times C$.

I wonder about the structure of maximal subgroups of $S$. Clearly, one of them is $C\times C$ and it is abelian.

I think the rest of the maximal subgroups is not abelian when $n\geq 2$. (When $n=1$ we have a dihedral group of order $8$.) However, if $M$ is a maximal subgroup of $S$ then $M$ has an abelian subgroup of index $2$.

Any remark and reference is welcome.

  • 1
    $\begingroup$ Do you mean wreathed? $\endgroup$ – J. W. Tanner Feb 4 at 16:14

Maximal subgroups of $p$-groups $G$ have index $p$ and contain $\Phi(G) = [G,G]G^p$.

In this example, with $a$ a generator of $C$, we have $\Phi(G) = \langle (a,a^{-1}), (a^2,1), (1,a^2) \rangle$ has index $4$ in $G$, and there are three maximal subgroups:

$\langle (a,1), (1,a) \rangle = C \times C$;

$\langle (a^2,1), (1,a^2), (a,a^{-1}), i \rangle$; and

$\langle (a^2,1), (1,a^2), (a,a^{-1}), (a,1)i \rangle$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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