1. Simple Abelian Group

Could someone explain why $\displaystyle G$ being simple and abelian implies it's cyclic?

Thanks!
-Chip

2. Originally Posted by chiph588@
Could someone explain why $\displaystyle G$ simple and abelian implies it is cyclic?

Thanks!
-Chip
choose $\displaystyle 1 \neq x \in G.$ then $\displaystyle H=\langle x \rangle$ is a non-trivial normal subgroup of $\displaystyle G$ because $\displaystyle G$ is abelian. so $\displaystyle H=G$ because $\displaystyle G$ is simple.

3. Originally Posted by NonCommAlg
choose $\displaystyle 1 \neq x \in G.$ then $\displaystyle H=\langle x \rangle$ is a non-trivial normal subgroup of $\displaystyle G$ because $\displaystyle G$ is abelian. so $\displaystyle H=G$ because $\displaystyle G$ is simple.
Ah yes this proof was rather... *takes glasses off* simple.

4. Originally Posted by chiph588@
Ah yes this proof was rather... *takes glasses off* simple.
G should be finite though.

5. Originally Posted by aliceinwonderland
G should be finite though.
of coures, infinite abelian group are never simple. also, obviously, the finite ones are either of order 1 or a prime number.