Conjugate of Subgroup is Subgroup

Theorem
Let $G$ be a group.

Let $H \le G$ be a subgroup of $G$.

Then the conjugate of $H$ by $a$ is a subgroup of $G$:


 * $\forall H \le G, a \in G: H^a \le G$

Proof
Let $H \le G$.

First, we show that $x, y \in H^a \implies x \circ y \in H^a$:

Next, we show that $x \in H^a \implies x^{-1} \in H^a$:

Thus by the Two-Step Subgroup Test, $H^a \le G$.