Constrained for a prime divisor implies not simple non-abelian

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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)) \le 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.