PreprintCC BY 4.0arXiv
On the order of a finite group with trivial centre and bounded conjugacy classes
Abstract
A group is called an n-bfc-group if each of its conjugacy classes is finite and contains at most n elements. By a theorem of B.~H.~Neumann the derived subgroup of such a group is finite of n-bounded order, whereas the order of the group itself need not be bounded, as extraspecial p-groups show. We prove that an n-bfc-group with trivial centre is finite of order at most n64(log n)5. Passing to the quotient by the hypercentre, we deduce that no assumption on the structure of the group is needed at all: for an arbitrary finite n-bfc-group G one has |G:Z∞(G)|<n64(log n)5. The finiteness of the group is required here only for the correct definition of the hypercentre.
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BibTeX
@misc{gorshkov2026order,
title = {{On the order of a finite group with trivial centre and bounded conjugacy classes}},
author = {Ilya Gorshkov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31387},
eprint = {2609.31387},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31387},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the order of a finite group with trivial centre and bounded conjugacy classes
AU - Ilya Gorshkov
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31387
UR - https://arxiv.org/abs/2609.31387
AB - A group is called an n-bfc-group if each of its conjugacy classes is finite and contains at most n elements. By a theorem of B.~H.~Neumann the derived subgroup of such a group is finite of n-bounded order, whereas the order of the group itself need not be bounded, as extraspecial p-groups show. We prove that an n-bfc-group with trivial centre is finite of order at most n^64(log n)^5. Passing to the quotient by the hypercentre, we deduce that no assumption on the structure of the group is needed at all: for an arbitrary finite n-bfc-group G one has |G:Z_∞(G)|<n^64(log n)^5. The finiteness of the group is required here only for the correct definition of the hypercentre.
ER -