Is a simple quotient group of finite $G$ isomorphic to any nontrivial simple normal subgroup of $G$? And if so, why?
To give context, assume that $G$ has the following two composition series:
$1 = H\unlhd I$ $\unlhd G$
and $1 = A\unlhd B$ $\unlhd G$.
I want to show that $G/I$ is isomorphic to $B$ without assuming the J-H Theorem.