Constrained for a prime divisor implies not simple non-abelian

Statement

Suppose $G$ is a finite group and $p$ is prime divisor of the order of $G$. Then, if $G$ is a p-constrained group, $G$ cannot be a simple non-abelian group.

Definitions used

p-constrained group

Further information: P-constrained group (?)

Suppose $G$ is a finite group and $P$ is a $p$-Sylow subgroup. We say that $G$ is $p$-constrained if we have:

$C_{G}(P\cap O_{p',p}(G))\leq O_{p',p}(G)$.

Facts used

1. Prime power order implies not centerless

Proof

We prove the contrapositive, which is somewhat easier.

Given: A finite simple non-abelian group $G$

To prove: $G$ is not $p$-constrained for any prime divisor $p$ of the order of $G$.

Proof: Since $p$ divides the order of $G$, we obtain that $O_{p',p}(G)$ is trivial, and hence $P\cap O_{p',p}(G)$ is trivial for any $p$-Sylow subgroup $P$ of $G$. We thus get:

$C_{G}(P\cap O_{p',p}(G))=G$

Since $O_{p',p}(G)$ is trivial, we see that the $p$-constraint condition is violated.