Any Subgroup of Index 2 in a Finite Group is Normal

Problem 16

Show that any subgroup of index $2$ in a group is a normal subgroup.

Contents

Hint.

1. Left (right) cosets partition the group into disjoint sets.
2. Consider both left and right cosets.

Proof.

Let $H$ be a subgroup of index $2$ in a group $G$.
Let $e \in G$ be the identity element of $G$.

To prove that $H$ is a normal subgroup, we want to show that for any $g\in G$, $gH=Hg$.
If $g \in H$, then this is true. So we assume that $g \not \in H$.

Note that left cosets partition $G$ into two disjoint sets since the index is $2$.
Since $g \not \in H$, these are $eH$ and $gH$. (If $gH=H$, then $g \in H$.)

Similarly right cosets partition $G$ into two disjoint sets.
These disjoint right cosets are $He$ and $Hg$.

Because of these partitions, we have as sets
$gH=G – eH=G-H=G-He=Hg.$ Therefore $H$ is a normal subgroup in $G$.

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