Show that $H$ is a normal subgroup of $G$ is it has index $2$.

Question

I'm stuck on this question for hours. I think I need a little help and a different point of view to complete it.

Let $H$ be a subgroup of a group $G$ and suppose that $H$ has index two in $G$, i.e., that there are precisely two elements in $G/H$. Show that $H$ is a normal subgroup of $G$.

Thanks in advance.

You will also find two proofs in the statement of this question. — Brian M. Scott, Nov 24 '13 at 20:45

score 1 · Answer 1 · answered Nov 24 '13 at 21:19

1

I have another proof of the folowing :

The number of left cosets of $H$ is equal to the numer of the right cosets. Now $G = H \cup aH= H \cup Ha$ (H has index 2), hence $aH=Ha$ and H is a normal subgroup.

answered Nov 24 '13 at 21:19

Eli Elizirov

1,830

Show that $H$ is a normal subgroup of $G$ is it has index $2$.

1 Answers1