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Updated Oct 8

Problem 117

claims/: The 1 claim page of Problem 117, one per claimant's result; the problem's standing derives from them.

Statement. Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.

Estimate $h(n)$ as well as possible.

Status. Open on the site (label OPEN; page last edited 23 January 2026). The site's proof-claims tab carries one full proof claim, submitted 18 August 2026 by Guillaume Lecomte and recorded as a pending claim on its claim page: that $\log_2 h(n)=n/2+o(n)$ , so $h(n)^{1/n}\to\sqrt{2}$ .

Source. erdosproblems.com/117, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #117, https://www.erdosproblems.com/117.

References.

[Er97f] Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. library card, chapter at pp. 1-10.
[Py87] Pyber, L., The number of pairwise noncommuting elements and the index of the centre in a finite group. J. London Math. Soc. (2) (1987), 287-295.

Formalization. None recorded.

Current assessment

The site's formulation asks to estimate $h(n)$ , the least number such that every group in which any $n+1$ elements include two distinct commuting ones is a union of at most $h(n)$ Abelian subgroups. The known bounds are exponential: Pyber [Py87] proved $c_1^n<h(n)<c_2^n$ for absolute constants $c_2>c_1>1$ , the upper bound through the theorem that a finite group with at most $n$ pairwise non-commuting elements has center of index at most $c^n$ , and the lower bound, which Erdős [Er97f] attributes to Isaacs, from extraspecial $2$ -groups; the library card is Pyber 1987. The base of the exponential was left open there.

One pending full claim, Lecomte 2026, asserts the sharp rate $\log_2 h(n)=n/2+O(\sqrt{n}(\log(n+2))^3)$ , with the lower bound from extraspecial $2$ -groups and the upper bound from a $p$ -group analysis through alternating forms, Sylow decomposition and a reduction to the centralizer of the derived subgroup. It is unreviewed and unpublished, so the derived standing is claimed, with claim value answered since the problem asks for an estimate. Should the claim be accepted, the question would be answered at the level of the exponential rate; the error term would remain a further estimate. The search behind this account covered the site's page, its discussion and proof-claims threads, Zenodo and arXiv on 2026-10-07.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (10)

Lecomte's sharp exponential rate for abelian covers problems/group_theory/E0117/claims Group Theory extremal_graph_theory/erdos_1997_some_unsolved_problems Group Theory Pyber: The Number of Pairwise Non-Commuting Elements and the Index of the Centre Corollary (p. 287): a group with n(G) = n is covered by c^n complete subgraphs of its commuting graph Example (p. 288): extraspecial groups of order 2^(2m+1) need at least 2^m+1 commuting subsets Lemma 3.1 (p. 288): every conjugacy class has at most 4n^2 elements Theorem (p. 287) and Theorem 6.1 (p. 294): the centre has index at most exponential in n(G)

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