PreprintCC BY 4.0arXiv
The Exact Central-Index Profile of Groups with Bounded Noncommutativity
Abstract
For a finite group G, let ω(G) be the largest size of a pairwise noncommuting set and put z(G)=[G:Z(G)]. We determine the maximum possible central index under ω(G)≤ k for every integer k≥1, giving an exact answer to Neumann's central-index question and sharpening Pyber's exponential upper bound. The maximum is $1$ for k=1,2, (k−1)2 for 3≤ k≤6, $81$ for k=7,8, and 4⌊(k−1)/2⌋ for k≥9. In the last range, with r=⌊(k−1)/2⌋, equality holds exactly when G'≅ C2 and G/Z(G)≅𝔽22r. We also obtain the sharp universal bound z(G)≤(81/64)2ω(G)−1. All maximizers are classified up to isoclinism. The results extend to arbitrary groups with finite clique number through finite isoclinic representatives.
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BibTeX
@misc{lecomte2026exact,
title = {{The Exact Central-Index Profile of Groups with Bounded Noncommutativity}},
author = {Guillaume Lecomte},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00943},
eprint = {2611.00943},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00943},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Exact Central-Index Profile of Groups with Bounded Noncommutativity
AU - Guillaume Lecomte
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00943
UR - https://arxiv.org/abs/2611.00943
AB - For a finite group G, let ω(G) be the largest size of a pairwise noncommuting set and put z(G)=[G:Z(G)]. We determine the maximum possible central index under ω(G)≤ k for every integer k≥1, giving an exact answer to Neumann's central-index question and sharpening Pyber's exponential upper bound. The maximum is $1$ for k=1,2, (k−1)^2 for 3≤ k≤6, $81$ for k=7,8, and 4^⌊(k−1)/2⌋ for k≥9. In the last range, with r=⌊(k−1)/2⌋, equality holds exactly when G'≅ C_2 and G/Z(G)≅𝔽_2^2r. We also obtain the sharp universal bound z(G)≤(81/64)2^ω(G)−1. All maximizers are classified up to isoclinism. The results extend to arbitrary groups with finite clique number through finite isoclinic representatives.
ER -