We 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.
Detomi E., Morigi M., Shumyatsky P. (2022). On the rank of a verbal subgroup of a finite group. JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 113(2), 145-159 [10.1017/S1446788721000069].
On the rank of a verbal subgroup of a finite group
Morigi M.;
2022
Abstract
We 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.| File | Dimensione | Formato | |
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