Hello,

I have a homework problem which asks:

Let G be a group and let H be a subgroup of G.

(a) Show that if H is index 2 in G then H is a normal subgroup of G.

(b) Show that if H is a normal subgroup of G then H is index 2 in G.

I proved the first part like so:

Assume H has an index of 2

Thus it has 2 left cosets which are forced to be H and xH = G/H

and the 2 right cosets are H and Hx = G/H

So, Hx = xH for all x

Thus H is normal.

However, I can't figure out how to go the other way.

Any help would be much appreciated.

Thanks.