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I'm stuck with a problem for an upcoming abstract algebra exam. Here it is:

Prove that if $N$ is a subgroup of a group $G$ and $[G\::\:N]=2$, then $N$ is normal in $G$.

I know that the fact that the index is two means that there are exactly two cosets of $N$ in $G$, but I'm stuck from there. Thanks for any help!

Sir_Math_Cat
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1 Answers1

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You just have to show that left cosets are the same as right cosets. Now there are two left cosets: $N$ and $aN$ for any $a\in N$, which constitute a partition of $G$. In other words $aN=G\setminus N$. For the same reason, the right coset $Na$ is $G\setminus N$. So they're equal.

Bernard
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