# $2$-power-torsion elements of a group

Let $$G$$ be a finite group, let $$P$$ be one of its $$2$$-sylow subgroups. Let $$H$$ be a proper subgroup of $$P$$, namely $$H with $$H\neq P$$. Is it possible that $$\bigcup_{g\in G}g^{-1}Hg=\bigcup_{g\in G}g^{-1}Pg?$$

• What about $G=A_4$ and $H$ a subgroup of order $2$. Then $P \lhd G$ and $\cup_{g \in G} g^{-1}Hg = P$. – Derek Holt Jan 6 at 20:54
• @Derek Right, thank you! – Qixiao Ma Jan 7 at 1:26
• Just to build on Derek Holt's example, there are many simple examples. For example, let $G$ be any of the simple groups ${\rm SL}(2,2^{n})$ where $n >1.$ Then $G$ has a unique conjugacy class of involutions, so f $P$ is a Sylow $2$-subgroup of $G$ and $T$ is a (proper) subgroup of $P$ of order $2,$ we have $\bigcup_{g \in G} g^{-1}Tg = \bigcup_{g \in G} g^{-1}Pg.$ – Geoff Robinson Jan 7 at 12:54