Index of 2 is equivalent to normal
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.