A generalization of Hall's theorem on hypercenter

2021/03/08 by Murashka, Viachaslau I., Vasil'ev, Alexander F.
#20F19 #FOS: Mathematics #Group Theory (math.GR) #Primary 20D25 #Secondary 20F17

paper · doi:10.48550/arxiv.2103.04900

Abstract

Let σ be a partition of the set of all primes and \mathfrakF be a hereditary formation. We described all formations \mathfrakF for which the \mathfrakF-hypercenter and the intersection of weak K-\mathfrakF-subnormalizers of all Sylow subgroups coincide in every group. In particular the formation of all σ-nilpotent groups has this property. With the help of our results we solve a particular case of L.A.~Shemetkov's problem about the intersection of \mathfrakF-maximal subgroups and the \mathfrakF-hypercenter. As corollaries we obtained P. Hall's and R. Baer's classical results about the hypercenter. We proved that the non-σ-nilpotent graph of a group is connected and its diameter is at most 3.

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