1. ## abstract algebra

1. Show that if E is normal in H, then f-1(E) is normal in G.
2. If G is cyclic, then f(G) is also cyclic.
Hint: G = <a>. Show: f(G)=< f(a)>.
3. I G is a cyclic group of infinite order then G is isomorphic to the additive group 2. Hint: f: G→Z by f(a^m) = m.

2. Originally Posted by vldo

1. Show that if E is normal in H, then f-1(E) is normal in G.
2. If G is cyclic, then f(G) is also cyclic.
Hint: G = <a>. Show: f(G)=< f(a)>.
3. I G is a cyclic group of infinite order then G is isomorphic to the additive group 2. Hint: f: G→Z by f(a^m) = m.
For 1, What is f-1(E)?

For 2, I can't really give much more of a hint than the hint given. What have you done so far?

For 3, you want to find an isomorphism between a group G=<a>, $a^n\neq 1$ for all $n \in \mathbb{N}$. What do you think this isomorphism is?

3. Originally Posted by vldo