Analogue of Thompson transitivity theorem fails for groups in which not every p-local subgroup is p-constrained

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Statement

It is possible to choose a finite group G and a prime number p such that G is not a group in which every p-local subgroup is p-constrained, and such that:

There is a subgroup A that is maximal among abelian normal subgroups in some p-Sylow subgroup of G such that the rank of A is at least three, and there is a prime q \ne p such that C_G(A) is not transitive on the set of maximal A-invariant q-subgroups.

In other words, the analogue of the Thompson transitivity theorem fails if we drop the assumption that the group is a group in which every p-local subgroup is p-constrained.

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References

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