Normal coverings of linear groups

TL;DR

This paper investigates the minimal number of conjugacy classes of proper subgroups needed to cover certain linear groups, providing bounds and explicit formulas for these covering numbers.

Contribution

It establishes new bounds and formulas for the covering number mma(G) of linear groups between oldsymbol{SL}_n(q) and oldsymbol{GL}_n(q), extending previous results.

Findings

oldsymbol{ ext{For } G ext{ with } ext{ extbf{SL}_n(q)} \u2264 G \u2264 extbf{GL}_n(q), ext{ bounds are } n/pi^2 < mma(G) \u2264 (n+1)/2.

oldsymbol{ ext{Explicit formulas for } mma(G) ext{ are derived in certain cases.}}

oldsymbol{ ext{Bounds improve understanding of subgroup coverings in linear groups.}}

Abstract

For a non-cyclic finite group $G$, let $\gamma(G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently established that $\gamma(S_n)$ and $\gamma(A_{n})$ are bounded above and below by linear functions of $n$. In this paper we show that if $G$ is in the range $\SL_{n}(q)\le G\le \GL_{n}(q)$ for $n>2$, then $n/\pi^2 < \gamma(G) \le (n+1)/2$. We give various alternative bounds, and derive explicit formulas for $\gamma(G)$ in some cases.