- #1
Rederick
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Homework Statement
If G is an abelian simple group then G is isomorphic to Zp for some prime p (do not assume G is a finite group).
Homework Equations
In class, we were told an example of a simple group is a cyclic group of prime order.
The Attempt at a Solution
Let G be an abelian simple group. Let a be an element in G that is not the identity. Let H be a cyclic subgroup of G, H=<a>. Since a is an element in G, it's non-trivial. Since G is abelian, H is normal in G. Since G is a simple group, it has only 2 normal subgroups, the identity and itself. Thus, H=G. Since G is cyclic and an abelian simple group, it has prime order. Hence G is isomorphic to Zp.
I feel uneasy about the last 2 sentences. Any advice to make this a more solid proof?