Lemma on containment in p'-core for Thompson transitivity theorem

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Statement

Suppose G is a finite group that is a group in which every p-local subgroup is p-constrained. Suppose A is maximal among abelian normal subgroups in a p-Sylow subgroup of G, and A is not cyclic, i.e., A \in SCN_2(p).

Suppose q is a prime number distinct from p, and Q is an A-invariant q-subgroup of G. Suppose H is a subgroup of G for which the p-core O_p(H) is nontrivial, and AQ \le H, then Q \le O_{p'}(H).

Related facts

Applications

Facts used

  1. Maximal among abelian normal subgroups in p-Sylow subgroup that is not cyclic implies every invariant p'-subgroup is in the p'-core in p-constrained group

References