PreprintCC BY 4.0arXiv
Mazurov's question on fixed points of automorphisms of free Burnside groups
Abstract
In the Kourovka Notebook, V.\,D.~Mazurov asked whether an automorphism α of prime order m of the infinite free Burnside group G=B(m,q) of prime period q that cyclically permutes the free generators of G must have a nontrivial fixed point. In this paper, we show that for m≥3 the answer is negative for all sufficiently large prime periods q. Precisely, we prove that for any odd m≥3, there exists qm>0 such that for every prime period q>qm, the automorphism of B(m,q) of order m that cyclically permutes its free generators is fixed-point-free.
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BibTeX
@misc{atabekyan2026mazurovs,
title = {{Mazurov's question on fixed points of automorphisms of free Burnside groups}},
author = {V. S. Atabekyan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33941},
eprint = {2609.33941},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33941},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mazurov's question on fixed points of automorphisms of free Burnside groups
AU - V. S. Atabekyan
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2609.33941
UR - https://arxiv.org/abs/2609.33941
AB - In the Kourovka Notebook, V.\,D.~Mazurov asked whether an automorphism α of prime order m of the infinite free Burnside group G=B(m,q) of prime period q that cyclically permutes the free generators of G must have a nontrivial fixed point. In this paper, we show that for m≥3 the answer is negative for all sufficiently large prime periods q. Precisely, we prove that for any odd m≥3, there exists q_m>0 such that for every prime period q>q_m, the automorphism of B(m,q) of order m that cyclically permutes its free generators is fixed-point-free.
ER -