Exercise 975Algebra › GroupsL1Added 3 Nov 2026

Subgroups of index 2 are normal

#index#normal#cosets

Statement

Show that any subgroup $H$ of index $2$ in $G$ is normal.

Hint

There are only two left cosets: $H$ and $G\setminus H$ . The same holds for right cosets.

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Solution

Cosets. Left cosets: $H$ and $gH$ for $g\notin H$ . Since $H$ has index $2$ , $gH=G\setminus H$ . Similarly, right cosets are $H$ and $Hg=G\setminus H$ .

Equality. For $g\notin H$ : $gH=G\setminus H=Hg$ . For $g\in H$ : $gH=H=Hg$ .

Conclusion. $gH=Hg$ for all $g$ , so $H\triangleleft G$ . Any index- $2$ subgroup is normal.

Subgroups of index 2 are normal

Statement

Solution

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