I have been going through some material on elementary group theory (Group Theory for Physicists, Christoph Ludeling) and in one place it states:
A subgroup which contains half of all elements, i.e. for which $|G|=2|H|$, is a normal subgroup.
Where a subgroup $H\subset G$ is called a normal subgroup if $gHg^{-1}=H~\forall g\in G$.
How would I go about proving this? Sorry if the answer exists somewhere, but I tried searching and could not find it.
Best Answer
The two left cosets of $H$ are $H$ and $G\setminus H$, and the two right cosets of $H$ are also $H$ and $G\setminus H$. Clearly $xH=Hx$ for every $x\in H$, and if $x\in G\setminus H$, then $xH=G\setminus H=Hx$ as well. Thus, $xH=Hx$ for each $x\in G$, and $H$ is normal in $G$.