## § Frobenius Kernel

#### § Some facts about conjugates of a subgroup

Let $H$ be a subgroup of $G$. Define $H_g \equiv \{ g h g^{-1} : h \in H \}$.
• We will always have $e \in H_g$ since $geg^{-1} = e \in H_g$.
• Pick $k_1 k_2 \in H_g$. This gives us $k_i = gh_ig^{-1}$. So, $k_1 k_2 = g h_1 g^{-1} g h_2 g^{-1} = g (h_1 h_2) g^{-1} \in H_g$.
• Thus, the conjugates of a subgroup is going to be another subgroup that has nontrivial intersection with the original subgroup.
• For inverse, send $k = ghg^{-1}$ to $k^{-1} = g h^{-1} g^{-1}$.