## A finite group with a cyclic Sylow 2-subgroup has a normal subgroup of maximal odd order

Prove that if $|G| = 2^nm$ where $m$ is odd and $G$ has a cyclic Sylow 2-subgroup then $G$ has a normal subgroup of order $m$.

We proceed by induction on $n$.

For the base case, suppose $|G| = 2m$ where $m$ is odd. Note that every Sylow 2-subgroup is cyclic. By this previous exercise, $G$ has a subgroup of index 2, hence order $m$, which is necessarily normal.

For the inductive step, suppose that for some $n \geq 1$ and all odd $m$, any group of order $2^nm$ which has a cyclic Sylow 2-subgroup has a normal subgroup of order $m$. Let $G$ be a group of order $2^{n+1}m$ and with a cyclic Sylow 2-subgroup. Let $x \in G$ be a generator of the Sylow 2-subgroup; i.e. $|x| = 2^{n+1}$. Let $\varphi : G \rightarrow S_G$ be the left regular representation of $G$; by this previous exercise, $\varphi(x)$ is an odd permutation. By this previous exercise, $G$ has a subgroup $H$ of index 2, which is necessarily normal in $G$. Note that all Sylow 2-subgroups of $G$ are cyclic, and that every Sylow 2-subgroup of $H$ is contained in a Sylow 2-subgroup of $G$. Thus $H$ has a cyclic Sylow 2-subgroup, in particular $H \cap \langle x \rangle = \langle x^2 \rangle$ since $\langle x \rangle$ has a unique subgroup of order $2^n$. By the induction hypothesis, $H$ has a normal subgroup $N$ of order $m$. Note that $H \leq N_G(N) \leq G$, and that since $H$ is maximal, $N_G(N) \in \{ H,G \}$.

We claim that $N$ is the unique subgroup of $H$ of order $m$. To prove this claim, suppose to the contrary that $M \leq H$ is a subgroup of order $m$. Now $NM \leq H$ is a subgroup and strictly contains $N$ and $M$. Moreover, we have $|NM| = |N| \cdot |M| /|M \cap N|$, so that by Lagrange, $|NM|$ is odd. This is a contradiction, though, because $m$ is the largest odd integer dividing $|H|$. Thus $N$ is the unique subgroup of order $m$. Hence $N$ is characteristic in $H$. By this previous exercise, then, $N$ is normal in $G$.

By induction, the conclusion holds for all groups of order $2^nm$ having a cyclic Sylow 2-subgroup.