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Finite groups with conjugacy classes of size 2pn

  • Antonio Beltran
  • Changguo shao
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Abstract

We extend Burnside's pa-Theorem on conjugacy classes of prime-power size by proving that no non-abelian finite simple group possesses a conjugacy class of size 2pn for any prime p and n≥ 0. Based on this, we partially extend a well-known result of L. Kazarin by showing that, given a finite group G, the subgroup generated by a conjugacy class xG of size 2pn where x has odd order, is a solvable normal subgroup of G. If instead the subgroup ⟨ xG ⟩ generated by such a class xG is non-solvable (which forces x to have even order), then p= 5 and ⟨ xG ⟩/S(⟨ xG ⟩) is isomorphic to the symmetric group S5, where S(⟨ xG ⟩) denotes the solvable radical of ⟨ xG ⟩.

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Cite

BibTeXRIS
BibTeX
@misc{beltran2026finite,
  title = {{Finite groups with conjugacy classes of size $2p^n$}},
  author = {Antonio Beltran and Changguo shao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18193},
  eprint = {2610.18193},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18193},
}
RIS
TY  - UNPB
TI  - Finite groups with conjugacy classes of size 2p^n
AU  - Antonio Beltran
AU  - Changguo shao
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18193
UR  - https://arxiv.org/abs/2610.18193
AB  - We extend Burnside's p^a-Theorem on conjugacy classes of prime-power size by proving that no non-abelian finite simple group possesses a conjugacy class of size 2p^n for any prime p and n≥ 0. Based on this, we partially extend a well-known result of L. Kazarin by showing that, given a finite group G, the subgroup generated by a conjugacy class x^G of size 2p^n where x has odd order, is a solvable normal subgroup of G. If instead the subgroup ⟨ x^G ⟩ generated by such a class x^G is non-solvable (which forces x to have even order), then p= 5 and ⟨ x^G ⟩/S(⟨ x^G ⟩) is isomorphic to the symmetric group S_5, where S(⟨ x^G ⟩) denotes the solvable radical of ⟨ x^G ⟩.
ER  -