# Subgroup of Index 2 contains all Squares of Group Elements

## Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$ whose index is $2$.

Then:

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

## Proof

By Subgroup of Index 2 is Normal, $H$ is normal in $G$.

Hence the quotient group $G / H$ exists.

Then we have:

 $\, \displaystyle \forall x \in G: \,$ $\displaystyle \paren {x^2} H$ $=$ $\displaystyle \paren {x H}^2$ $\displaystyle$ $=$ $\displaystyle H$ as $G / H$ is of order $2$

$\blacksquare$