Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 8 down vote accepted

If $x\not\in H$, then $x^2$ is in either $H$ or $xH$. If $x^2\in xH$ then $x^2=xh$ for some $h\in H$. But then $x=h\in H$.

share|cite|improve this answer
Wonderful. Now I got it! Thank you very much! – William T. Nov 22 '11 at 15:41

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.