# Sylow 2-subgroups in non-abelian group of order 2012

Let $G$ be a group of order 2012.

I have shown the following:

• $G$ has a unique normal subgroup $S$ of order 503
• $S$ is cyclic and the automorphism group $\textrm{Aut}(S)$ contains an unique element of order 2, namely the one sending $x$ to $x^{-1}$.
• If $H$ is a Sylow 2-subgroup of $G$, then $G/S\simeq H$ and $G=SH$.

I am then asked to show that if $G$ is not abelian, then the Sylow 2-subgroups are isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2$. Any suggestions on how to do that?

• @DanielFischer They would have to be $\mathbb{Z}_4$. But where to I go from there? – Student G Jun 5 '14 at 20:35
• @StudentG: you've been asked the impossible. It is perfectly ok for the Sylow 2-subgroups to be cyclic. – Jack Schmidt Jun 5 '14 at 20:37
• @JackSchmidt I have? Could you provide a counter example? – Student G Jun 5 '14 at 20:41
• Take a semidirect product. – Daniel Fischer Jun 5 '14 at 20:44
• OK. Thanks to both of you. – Student G Jun 5 '14 at 20:56

Here is a counterexample: Consider the matrix group defined over the integers mod 503: $$G = \left\langle \left[ \begin{smallmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \\ \end{smallmatrix} \right], \left[ \begin{smallmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{smallmatrix} \right] \right\rangle$$