Bounded-Generator Expander Families of Finite Groups

Definitions and mathematical statement

Let $(G_i)_{i\in I}$ be the infinite family of finite groups in the source. Under the usual undirected Cayley-graph convention choose symmetric generating sets $S_i=S_i^{-1}\subseteq G_i\setminus\{1\}$, and form $\operatorname{Cay}(G_i,S_i)$ with edges between $g$ and $gs$. The task asks whether the given family admits constants $D<\infty$, $h>0$ and such sets $S_i$ with \[ |S_i|\leq D\quad\text{for all }i, \] \[ |\partial_E A|\geq h|A| \quad\text{for every }i\text{ and }\varnothing\ne A\subseteq G_i \text{ with }|A|\leq|G_i|/2. \] Here $\partial_E A$ consists of graph edges with exactly one endpoint in $A$. The constants are fixed for the family, while the generating set may depend on $i$. This is the standard bounded-degree edge-expansion formulation; normalization by degree or a compatible spectral-gap formulation needs its graph conventions fixed, but no particular generating set must work for every group.

A uniform bound on the number of abstract generators is a necessary prerequisite; the source does not assume it or give other structural information about $(G_i)$. Thus the wording is a family-dependent existence/classification problem, not a supported affirmative conjecture for every arbitrary sequence of finite groups. Even uniform generation alone is not asserted here to imply expansion. The question does not allow discarding members to obtain a favorable subfamily unless that is separately specified.

The record says infinite family, without explicitly requiring $|G_i|\to\infty$ or unbounded order. Such growth is usually part of the nontrivial expander-family setting; infinitely repeated copies of bounded groups need a convention. No distinctness or growth requirement is silently appended.

The literature triage checked 2026-08-21 reports uniform bounded-generator expanders for all nonabelian finite simple groups, including the formerly exceptional Suzuki groups, while retaining the broader family-dependent question. That supplied special-family result is not extended to arbitrary groups. Its partially-solved label and wording-verification note remain dated metadata, not independently checked current conclusions.

Short English statement

For the given finite-group family, can one choose generators with one common size bound and one common positive expansion constant? The sets may vary by group. The source supplies no family hypotheses or growth convention, so this is not a universal affirmative claim about arbitrary finite groups.

Sources

• Local UnsolvedMath registry, permanent ID 30000806, source code catalogue entry, "Bounded-Generator Expander Families of Finite Groups," including the unchanged family-existence question and special simple-group progress.
• Permutation Groups (2008), Oberwolfach Reports, pp. 2173--2236, DOI 10.4171/owr/2007/37: https://doi.org/10.4171/owr/2007/37. The citation uses 2008, while the DOI and proposed-year metadata use 2007.
• Supplied reference "Kassabov--Lubotzky--Nikolov, finite simple groups as expanders": https://doi.org/10.1073/pnas.0510337103.
• Supplied reference "Suzuki groups as expanders": https://arxiv.org/abs/1005.0782.
• Literature triage checked 2026-08-21; record updated 2026-09-16. No external source has been visited or independently checked here.