# Product of Conjugates equals Conjugate of Products

## Theorem

Let $\struct {G, \circ}$ be a group.

Then:

$\forall a, x, y \in G: \paren {a \circ x \circ a^{-1} } \circ \paren {a \circ y \circ a^{-1} } = a \circ \paren {x \circ y} \circ a^{-1}$

That is, the product of conjugates is equal to the conjugate of the product.

## Proof

Follows directly from the group axioms.

$\blacksquare$