Odd order matrix in $GL_n(\mathbb F_2)$ that doesn't commute with any order $2$ matrix? Is it the case that for all $n$, there is an invertible matrix over $\mathbb F_2$ of odd order which does not commute with any matrix of order $2$ in $GL_n(\mathbb F_2)$. I think this is equivalent to asking if there is a maximal cyclic subgroup of odd order.
Edit: Undeleted since now I can ask questions about Putnam.

Let $\alpha$ be the multiplication with a generator of $\mathbb{F}_{2^n}^\times$ and $\tau$ an invertible $\mathbb{F}_2$-linear map of order $2$ commuting with $\alpha$ (hence $\alpha^l\tau = \tau\alpha^l$ for all $l\in \mathbb{Z}$). Take $t$ such that $\tau 1=\alpha^t$.
For $v\ne0$ there exists an $l$ such that $v=\alpha^l$. Therefore $\tau v = \tau\alpha^l 1=\alpha^l\tau1=\alpha^l\alpha^t=\alpha^t\alpha^l=\alpha^tv$ showing that $\tau$ is multiplication with $\alpha^t$ (as $v$ was arbitrary). As $\tau$ has order $2$, it has to be multiplication with $-1$, but $\mathbb{F}_2$ has characteristic $2$.
