Are all simple groups solvable?

Question

I came across a proof where it is assumed that a simple group is solvable, but I can't really understand how.

Is this true? If yes, help to prove it would be very helpful.

PS : Question was to prove that all groups of order < 50 are solvable.

Answer 1

No, if a simple group is solvable it must be abelian. Because since it has no normal groups it won't be possible to find a factor group.

The smallest non-abelian simple group is $A_5$ and it has order $60$, so every simple group of order less than $50$ is in fact abelian.

Answer 2

If $G$ is a solvable group then there exists a finite chain of subgroups $$G\geq G^1\geq G^2\geq...\geq G^n\geq1$$ such that $G^{i+1}=[G^i, G^i]$.

Now, $[G, G]$ is a normal subgroup of $G$ (why?), and so if $G$ is non-trivial and simple then either $[G, G]=G$ or $[G, G]=1$. In the first case, $G$ is not solvable (as $G^i=G$ for all $i$). In the second case, $G$ is abelian (why?). Therefore,

A simple group is solvable if and only if it is abelian.