Local subgroup of finite group is contained in p-local subgroup for some prime p

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Statement

Suppose G is a finite group and H is a local subgroup of G, i.e., it is the normalizer of some nontrivial solvable subgroup of G. Then, there exists a prime number p and a p-local subgroup K of G such that H \le K.

Proof

Proof idea

Suppose H = N_G(Q) for a solvable subgroup Q of G. The idea is to find a prime p such that O_p(Q) (the p-core of Q) is nontrivial. Let K = N_G(O_p(Q)). We show that H \le K, completing the proof.