Rank properties of commutators in finite groups
Abstract
For a subset S of a finite group G, let IG(S) denote the set of commutators [g,x]=g−1gx, where g∈ G and x∈ S. Suppose that a finite group G has a Carter subgroup C, that is, a nilpotent subgroup containing its normalizer. Suppose that any subgroup generated by a subset of IG(C) is r-generated. We prove that if G is soluble, then the derived subgroup G' has r-bounded rank. We produce examples showing that the solubility condition cannot be dropped. For any finite group, we prove that the rank of G' is (r,l)-bounded, where l is the maximum rank of composition factors of G isomorphic to PSL2(q) for q≡ 7 (mod8). We also prove in the general case that G' has r-bounded rank under the additional condition that for any x∈ IG(C), any subgroup generated by a subset of IG(x) is r-generated. The proofs rely on the classification of finite simple groups, using which we prove that if a finite simple group G has an element x of prime order p such that any subgroup generated by a subset of IG(x) is r-generated, then G has r-bounded (Prüfer) rank or is isomorphic to PSL2(q) with q ≡ 3 (mod4) when p=2, or to PSL2(q) with q ≡ −1 (mod p) when p≠ 2.
Related
- The Conjugacy Problem for Recursively Presented Virtually Soluble Groups of Finite Prüfer Rank
- Generalized Erdős--Ginzburg--Ziv Constants for finite abelian groups of rank three
- Infinite girth and profinite properties of residually finite groups
- Equality cases for matrix spaces with bounded-rank commutators
- Failure of the congruence Hasse principle for commutators in orientable surface groups
BibTeX
@misc{acciarri2026rank,
title = {{Rank properties of commutators in finite groups}},
author = {Cristina Acciarri and Robert M. Guralnick and Evgeny Khukhro and Pavel Shumyatsky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28309},
eprint = {2609.28309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28309},
}
RIS
TY - UNPB
TI - Rank properties of commutators in finite groups
AU - Cristina Acciarri
AU - Robert M. Guralnick
AU - Evgeny Khukhro
AU - Pavel Shumyatsky
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28309
UR - https://arxiv.org/abs/2609.28309
AB - For a subset S of a finite group G, let I_G(S) denote the set of commutators [g,x]=g^−1g^x, where g∈ G and x∈ S. Suppose that a finite group G has a Carter subgroup C, that is, a nilpotent subgroup containing its normalizer. Suppose that any subgroup generated by a subset of I_G(C) is r-generated. We prove that if G is soluble, then the derived subgroup G' has r-bounded rank. We produce examples showing that the solubility condition cannot be dropped. For any finite group, we prove that the rank of G' is (r,l)-bounded, where l is the maximum rank of composition factors of G isomorphic to PSL_2(q) for q≡ 7 (mod8). We also prove in the general case that G' has r-bounded rank under the additional condition that for any x∈ I_G(C), any subgroup generated by a subset of I_G(x) is r-generated. The proofs rely on the classification of finite simple groups, using which we prove that if a finite simple group G has an element x of prime order p such that any subgroup generated by a subset of I_G(x) is r-generated, then G has r-bounded (Prüfer) rank or is isomorphic to PSL_2(q) with q ≡ 3 (mod4) when p=2, or to PSL_2(q) with q ≡ −1 (mod p) when p≠ 2.
ER -