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PreprintCC BY 4.0arXiv

Regular automorphisms of periodic p'-groups need not be splitting

  • V. S. Atabekyn
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Abstract

For a fixed prime p, a group is called a p'-group if the orders of all its elements are coprime to p. It is well known that in a finite p'-group an automorphism of order p is regular if and only if it is a splitting automorphism of period p. In the Kourovka Notebook, V.\,D.~Mazurov asked whether these two notions coincide for periodic p'-groups. We answer this question in the negative by constructing a periodic group of prime period q that admits a regular automorphism of order p ≠ q which is not splitting.

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Cite

BibTeXRIS
BibTeX
@misc{atabekyn2026regular,
  title = {{Regular automorphisms of periodic $p'$-groups need not be splitting}},
  author = {V. S. Atabekyn},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15360},
  eprint = {2610.15360},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15360},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Regular automorphisms of periodic p'-groups need not be splitting
AU  - V. S. Atabekyn
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15360
UR  - https://arxiv.org/abs/2610.15360
AB  - For a fixed prime p, a group is called a p'-group if the orders of all its elements are coprime to p. It is well known that in a finite p'-group an automorphism of order p is regular if and only if it is a splitting automorphism of period p. In the Kourovka Notebook, V.\,D.~Mazurov asked whether these two notions coincide for periodic p'-groups. We answer this question in the negative by constructing a periodic group of prime period q that admits a regular automorphism of order p ≠ q which is not splitting.
ER  -