I want to know a proof of this fact: "every simple group has a minimal simple group as a subquotient." (If $H$ and $K$ are two subgroups of $G$ s.t. $H\lhd K$, then $\frac{K}{H}$ is called a subquotient of $G$) many thanks.
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
If every proper subgroup of G is soluble, then it is done. So there is a nonsoluble subgroup H with every subgroup soluble. Let K be maximal normal subgroup of H, then H/K is a minimal simple group. 

