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

## 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 $Q \le O_{p'}(H)$.

## 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