Hello!

I have some problems with the following exercise:

Let G be a noncommutative group of order 12, H be a 3-Sylow group in G, and let T:G->{maps G/H->G/H}, where T(g) is defined as the map which sends aH -> gaH.

Show that:

1) T is not injective, if and only if H is normal in G

2) Use 1) to prove that if H is not normal in G, then G is isomorphic to the alternating group A_4.

Does anyone have an idea for 1) or 2)?

Marco