Kourovka Notebook Problem 20.17

Definitions and mathematical statement

For a finite non-cyclic group $G$, its normal covering number is \[ \gamma(G)=\min\left\{r\in{\mathbb{N}}:\ r\geq1,\ \exists H_1,\ldots,H_r<G,\quad G=\bigcup_{i=1}^{r}\ \bigcup_{g\in G}gH_i g^{-1}\right\}, \] where $H_i<G$ means that $H_i$ is a proper subgroup. Let $S_n$ and $A_n$ denote the symmetric and alternating groups on $n$ letters. All limits below are taken as $n\to\infty$ through the indicated parity. We may restrict to $n\geq4$, so both groups are non-cyclic; discarding finitely many indices does not change these limits.

The original questions of D. Bubboloni, C. E. Praeger, and P. Spiga are to determine the following exact values: \begin{align*} \text{(ae)}\quad&\liminf_{\substack{n\to\infty\\n\text{ even}}}\frac{\gamma(S_n)}{n},\\ \text{(ao)}\quad&\liminf_{\substack{n\to\infty\\n\text{ odd}}}\frac{\gamma(S_n)}{n},\\ \text{(b)}\quad&\liminf_{\substack{n\to\infty\\n\text{ even}}}\frac{\gamma(A_n)}{n}\quad\text{and}\quad\liminf_{\substack{n\to\infty\\n\text{ odd}}}\frac{\gamma(A_n)}{n},\\ \text{(c)}\quad&\limsup_{\substack{n\to\infty\\n\text{ even}}}\frac{\gamma(S_n)}{n},\\ \text{(d)}\quad&\limsup_{\substack{n\to\infty\\n\text{ odd}}}\frac{\gamma(A_n)}{n}\quad\text{and}\quad\limsup_{\substack{n\to\infty\\n\text{ even}}}\frac{\gamma(A_n)}{n}. \end{align*} The supplied record labels the overall problem partially solved. Its background reports that parts (ae), (c), and (d) are solved, with respective values \[ \liminf_{\substack{n\to\infty\\n\text{ even}}}\frac{\gamma(S_n)}n=\frac16, \qquad \limsup_{\substack{n\to\infty\\n\text{ even}}}\frac{\gamma(S_n)}n=\frac14, \] \[ \limsup_{\substack{n\to\infty\\n\text{ odd}}}\frac{\gamma(A_n)}n=\frac13, \qquad \limsup_{\substack{n\to\infty\\n\text{ even}}}\frac{\gamma(A_n)}n=\frac14. \] These reported resolutions are attributed to S. Eberhard and C. Mellon (2025). The remaining exact-value questions in the supplied record are (ao) and both quantities in (b). Its comments of 2025, attributed to the same reference, report \[ \frac1{12}\leq\liminf_{\substack{n\to\infty\\n\text{ odd}}}\frac{\gamma(S_n)}n\leq\frac16, \] \[ \frac1{18}\leq\liminf_{\substack{n\to\infty\\n\text{ even}}}\frac{\gamma(A_n)}n\leq\frac16, \qquad \frac16\leq\liminf_{\substack{n\to\infty\\n\text{ odd}}}\frac{\gamma(A_n)}n\leq\frac4{15}. \] The statement also reports as known that \[ \limsup_{\substack{n\to\infty\\n\text{ odd}}}\frac{\gamma(S_n)}n=\frac12. \] All status labels, bounds, and reported equalities here reproduce supplied source assessments; they have not been independently verified. The source gives no separate citation for the last equality, and its status-review note is truncated, although the complete supplied background gives all reported values for part (d).

Short English statement

Determine the exact asymptotic lower and upper limits, separately for even and odd degrees, of the normal covering numbers of symmetric and alternating groups divided by their degrees. The supplied record reports solutions to (ae), (c), and (d), leaving the exact lower limits in (ao) and (b) unresolved, with the stated 2025 bounds. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701675$, exact source code catalogue entry, title Kourovka Notebook Problem 20.17. The supplied status and canonical status are partially_solved.
• D. Bubboloni, C. E. Praeger, and P. Spiga, problem 20.17, proposed in 2022, in The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022); the supplied record also cites the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• S. Eberhard and C. Mellon, Bull. London Math. Soc. (2025), https://doi.org/10.1112/blms.70154. The supplied record attributes the reported resolutions of (ae), (c), and (d), and the comments of 2025 concerning bounds in (ao) and (b), to this reference. No article title is supplied.
• The supplied status review has check date 2026-10-16, evidence label maintained_source, and refers to the 21st issue of the Kourovka Notebook (October 2026), arXiv:1401.0300v48. This is supplied dated curation, not an independently verified literature assessment. Its notes field ends mid-sentence during the discussion of part (d); the complete background separately supplies both values.