Power of Coset Product is Coset of Power

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

Let $N$ be a normal subgroup of $G$.

Let $a \in G$.

Then:


 * $\forall n \in \Z: \paren {a \circ N} = \paren {a^n} \circ N$

Proof
From Quotient Group is Group, the operation:
 * $\forall a, b \in G: \paren {a \circ N} \circ \paren {b \circ N} = \paren {a \circ b} \circ N$

is the group product in the quotient group $\struct {G / N, \circ}$.

The result follows directly by definition of power of group element.