I'm having some trouble understanding cosets.
If I understand it correctly, cosets form a partition. So, if I have $|G:H| = 2$, then $G = H \cup xH$. Right?
In an exercise, I'm asked the following:
"For $G$ group and $H \le G$ such that $|G:H| = 2$, prove that $x^2 \in H$ for any $x \in G$"
If $x \in H$, then $x^2 \in H$ because $H$ is a group. But, if $x \notin H$, then $x \in gH$ for any $g \in G$, right? Then $x^2$ = $(gh)^2$ but I can't see why this is in $H$.
Can you give me some hint? Thanks for stopping by.