Large orbits in coprime actions of solvable groups

Dolfi, Silvio

doi:10.1090/S0002-9947-07-04155-4

Large orbits in coprime actions of solvable groups
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Trans. Amer. Math. Soc. 360 (2008), 135-152 Request permission

Abstract:

Let $G$ be a solvable group of automorphisms of a finite group $K$. If $|G|$ and $|K|$ are coprime, then there exists an orbit of $G$ on $K$ of size at least $\sqrt {|G|}$. It is also proved that in a $\pi$-solvable group, the largest normal $\pi$-subgroup is the intersection of at most three Hall $\pi$-subgroups.

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