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PreprintCC BY 4.0arXiv

Finite groups with large power-avoiding subsets

  • Simon R. Blackburn
  • Sarah B. Hart
  • Daniel McVeagh
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Abstract

A subset X of a finite group G is k-power-avoiding if for all g∈ G we have that {g,gk}not⊆ X. The paper shows that if G contains a k-power-avoiding subset X with |X|≥ |G|−c, then the group Gk generated by the kth powers of elements of G has bounded order (in k and c). We provide more detailed structural results when c≤ 2, and in particular we classify the groups which arise when c≤ 2 and k is prime.

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Cite

BibTeXRIS
BibTeX
@misc{blackburn2026finite,
  title = {{Finite groups with large power-avoiding subsets}},
  author = {Simon R. Blackburn and Sarah B. Hart and Daniel McVeagh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18513},
  eprint = {2609.18513},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18513},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finite groups with large power-avoiding subsets
AU  - Simon R. Blackburn
AU  - Sarah B. Hart
AU  - Daniel McVeagh
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18513
UR  - https://arxiv.org/abs/2609.18513
AB  - A subset X of a finite group G is k-power-avoiding if for all g∈ G we have that {g,g^k}not⊆ X. The paper shows that if G contains a k-power-avoiding subset X with |X|≥ |G|−c, then the group G^k generated by the kth powers of elements of G has bounded order (in k and c). We provide more detailed structural results when c≤ 2, and in particular we classify the groups which arise when c≤ 2 and k is prime.
ER  -