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On Shalev's Quantitative Version of Thompson's Conjecture

  • Ohad Sheinfeld
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Abstract

The classical Thompson's conjecture asserts that in any finite nonabelian simple group G, there exists a conjugacy class S whose square is equal to the entire group. Shalev (Annals Math., 2009) conjectured that a much stronger quantitative version holds: there exists ε> 0 such that for any conjugacy class S with |S| ≥ |G|1−ε, we have G ∖ {1} ⊆ S2. While the conjecture turns out to fail in general, we prove a close variant of the conjecture over PSLn(2): There exist ε>0 and N ∈ ℕ such that for all n≥ N, every conjugacy class S⊆PSLn(2) with |S|≥|PSLn(2)|1−ε satisfies {x∈PSLn(2):supp(x)≠1}⊆ SS−1. On the other hand, we show that for each ε>0, there exist conjugacy classes S ⊂ PSLn(8) of size >|PSLn(8))|1−ε for which SS−1 misses even elements with support size linear in n. In addition, for every odd n ≥ 5, we construct an inverse-closed normal set in PSLn(5) with fixed positive density whose square misses a nonidentity element. This answers on the negative a question of Shalev (ECM, 2021) and disproves a conjecture of Pyber (Kourovka notebook, 2022).

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Cite

BibTeXRIS
BibTeX
@misc{sheinfeld2026shalevs,
  title = {{On Shalev's Quantitative Version of Thompson's Conjecture}},
  author = {Ohad Sheinfeld},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22004},
  eprint = {2609.22004},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22004},
}
RIS
TY  - UNPB
TI  - On Shalev's Quantitative Version of Thompson's Conjecture
AU  - Ohad Sheinfeld
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.22004
UR  - https://arxiv.org/abs/2609.22004
AB  - The classical Thompson's conjecture asserts that in any finite nonabelian simple group G, there exists a conjugacy class S whose square is equal to the entire group. Shalev (Annals Math., 2009) conjectured that a much stronger quantitative version holds: there exists ε> 0 such that for any conjugacy class S with |S| ≥ |G|^1−ε, we have G ∖ {1} ⊆ S^2. While the conjecture turns out to fail in general, we prove a close variant of the conjecture over PSL_n(2): There exist ε>0 and N ∈ ℕ such that for all n≥ N, every conjugacy class S⊆PSL_n(2) with |S|≥|PSL_n(2)|^1−ε satisfies {x∈PSL_n(2):supp(x)≠1}⊆ SS^−1. On the other hand, we show that for each ε>0, there exist conjugacy classes S ⊂ PSL_n(8) of size >|PSL_n(8))|^1−ε for which SS^−1 misses even elements with support size linear in n. In addition, for every odd n ≥ 5, we construct an inverse-closed normal set in PSL_n(5) with fixed positive density whose square misses a nonidentity element. This answers on the negative a question of Shalev (ECM, 2021) and disproves a conjecture of Pyber (Kourovka notebook, 2022).
ER  -