TITLE: A subgroup with an index of 2 is normal to the group.
QUESTION [7 upvotes]: Why is a subgroup with the index of 2 normal to the group?  I believe it has something to do with it being half and having nowhere to go.

REPLY [8 votes]: Well, you're right; it's about "being half and having nowhere to go."
So let $G$ be a group and let $H$ be a subgroup of $G$ such that $[G:H] = 2$; then $H$ has two cosets of the form $gH$ or $Hg$ in $G$; it has two left cosets, and it has  two right cosets.  Now for any $g \in G$, $g \notin H$, we have $G = H \cup gH = H \cup Hg$ as the coset partitions of $G$.  Since $H \cap gH = H \cap Hg = \varnothing$, we must have $gH = Hg$ since clearly $gH \ne H \ne Hg$; remember $g \notin H$ here.  But $gH = Hg$ implies $gHg^{-1} = H$, whence $H$ is normal in $G$.
Hope this helps.  Holiday Cheer,
and of course,
Fiat Lux!!!