Dear Friends

Could someone help me in proving the following:

Suppose H is a subgroup of G of odd order and that |H|=5. Show that H is a subset of G.

What I tried is the following:

Since |H|=5 then H is cyclic so H=<h>={e, h, h^2,h^3,h^4}. For g in G (g^-1)*h*g is in H as H is normal so (g^-1)*h*g=h^k where k=0,1,2,3,4 e is the identity element.

If K=0 the we have h=e which is a contardiction as h is a generator.

If k=1 then we are done.

For the other cases I am not able to deal with them.

G has odd order implies that g and g^-1 are distinct. What can I also get from this infromation?

Regards.