For a) I proved the subgroup property. My problem is the example. H clearly has to be a group of some kind else the statement "H is not a subgroup of N_G(H)" is empty of meaning. So H has to be a group, but not a subgroup of G. I can't image how this could arise. Any hints?Let H be a subgroup of a group G.

a) Show that $\displaystyle H \leq N_G(H)$. Give an example to show that this is not necessarily true if H is not a subgroup of G.

b) Show that $\displaystyle H \leq C_G(H)$ if and only if H is abelian.

For b) I have proven that if H is abelian then $\displaystyle H \leq C_G(H)$. I am having trouble with "If $\displaystyle H \leq C_G(H)$ implies H is abelian. If $\displaystyle C_G(H)$ were abelian it would be easy, but I don't think this is true.

Thanks for any help!

-Dan