Subgroup Containing all Squares of Group Elements is Normal/Corollary

Corollary to Subgroup Containing all Squares of Group Elements is Normal

Let $G$ be a group.

Let $H$ be a subgroup of $G$ with the property that:

$\forall x \in G: x^2 \in H$

The quotient group $G / H$ is abelian.

Proof

From Subgroup Containing all Squares of Group Elements is Normal, $H$ is normal in $G$.

For every $x \in G$, we have:

\(\ds \paren {x H}^2\)

\(=\)

\(\ds x^2 H\)

\(\ds \)

\(=\)

\(\ds H\)

as $x^2 \in H$

Thus by definition $G / H$ is boolean.

The result follows by Boolean Group is Abelian.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $13$