Prove that if there exists a chain of subgroups such that and each is simple then is simple.

Suppose we have such a chain, and let be a normal subgroup. Note that is a normal subgroup for each . Since each is simple, we have .

Suppose for all . Then . Hence .

Suppose now that for some . Note that if for some , then . Thus if we let be minimal such that , then by induction for all . Then . Thus .

Since every normal subgroup of is trivial, is simple.

## Comments

Same proof as yours, but less latex

http://nhanttruong.wordpress.com/2011/07/08/a-simple-chain/

Thanks!