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20.17 (2022)

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The normal covering number $\gamma(G)$ of a finite non-cyclic group $G$ is the minimum number of proper subgroups of $G$ such that $G$ is the union of their conjugates.
$\qquad$ (ae) What is the exact value of $\lim \inf_{n \text{ even}} \gamma(S_n)/n$?
$\qquad$ (ao) What is the exact value of $\lim \inf_{n \text{ odd}} \gamma(S_n)/n$?
$\qquad$ (b) What are the exact values of $\lim \inf_{n \text{ even}} \gamma(A_n)/n$ and $\lim \inf_{n \text{ odd}} \gamma(A_n)/n$?
$\qquad$ (c) What is the exact value of $\lim \sup_{n \text{ even}} \gamma(S_n)/n$? It is known that $\lim \sup_{n \text{ odd}} \gamma(S_n)/n = 1/2$.
$\qquad$ (d) What are the exact values of $\lim \sup_{n \text{ odd}} \gamma(A_n)/n$ and $\lim \sup_{n \text{ even}} \gamma(A_n)/n$?

Contributor: D. Bubboloni, C. E. Praeger, P. Spiga

Progress

*(ae) It is 1/6 (S. Eberhard, C. Mellon, Bull. London Math. Soc. (2025), https://doi.org/10.1112/blms.70154).
(b) Comment of 2025: These belong to $[1/18, 1/6]$ for $n$ even, and to $[1/6, 4/15]$ for $n$ odd (S. Eberhard, C. Mellon, Bull. London Math. Soc. (2025), https://doi.org/10.1112/blms.70154).
*(c) It is 1/4 (S. Eberhard, C. Mellon, Bull. London Math. Soc. (2025), https://doi.org/10.1112/blms.70154).
*(d) These are 1/3 for the odd case, and 1/4 for the even case (S. Eberhard, C. Mellon, Bull. London Math. Soc. (2025), https://doi.org/10.1112/blms.70154).

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