A Universe of Sorts

§ 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 $k^{-1} = g h^{-1} g^{-1}$ .

§ Frobenius groups