Condition for Power of Element of Quotient Group to be Identity

Theorem
Let $\struct G$ be a group whose identity is $e$.

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

Let $a \in G$.

Then:
 * $\paren {a N}^n$ is the identity of the quotient group $G / N$


 * $a^n = e$
 * $a^n = e$

Necessary Condition
Let $\paren {a N}^n$ be the identity of $G / N$.

Then:

Sufficient Condition
Let $a^n = e$.

Then:

By Quotient Group is Group, $N$ is the identity of $G / N$.