ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

Pavel Shumyatsky,Marta Morigi,Eloisa Detomi

doi:10.1017/s1446788721000069

Pavel Shumyatsky, Marta Morigi + Show 1 more

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https://doi.org/10.1017/s1446788721000069

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Abstract

AbstractWe show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

Highlights

Guralnick [11] and Lucchini [17] independently proved that if all Sylow subgroups of a finite group G can be generated by d elements, the group G itself can be generated by d + 1 elements
In the present paper we are concerned with the question whether the rank of a verbal subgroup w(G) can be bounded in terms of ranks of nilpotent subgroups generated by w-values
Let w be a multilinear commutator word and G a soluble or simple finite group in which every nilpotent subgroup generated by w-values has rank at most r

Summary

Introduction

Guralnick [11] and Lucchini [17] independently proved that if all Sylow subgroups of a finite group G can be generated by d elements, the group G itself can be generated by d + 1 elements. Let w be a group-word and G a finite group in which every nilpotent subgroup generated by w-values has rank at most r. Let w be a multilinear commutator word and G a soluble or simple finite group in which every nilpotent subgroup generated by w-values has rank at most r.

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Conclusion