Normal coverings of linear groups
John R. Britnell, Attila Maroti

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 , let denote the smallest number of conjugacy classes of proper subgroups of needed to cover . Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently established that and are bounded above and below by linear functions of . In this paper we show that if is in the range for , then . We give various alternative bounds, and derive explicit formulas for in some cases.
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Taxonomy
TopicsFinite Group Theory Research · Coding theory and cryptography · Limits and Structures in Graph Theory
