Condition for Power of Element of Quotient Group to be Identity

Theorem

Let $G$ 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 $G / N$

if and only if:

$a^n \in N$

Proof

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

Then:

\(\ds \paren {a N}^n\)

\(=\)

\(\ds N\)

Quotient Group is Group: Group Axiom $\text G 2$: Existence of Identity Element

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {a^n} N\)

\(=\)

\(\ds N\)

Power of Coset Product is Coset of Power

\(\ds \leadstoandfrom \ \ \)

\(\ds a^n\)

\(\in\)

\(\ds N\)

Coset Equals Subgroup iff Element in Subgroup

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $13$