Subgroup coverings of some linear groups

Abstract

Dedicated to Bernhard Neumann on his 90th birthday.A cover for a group is a collection of proper subgroups whose union is the whole group. A cover is minimal if no other cover contains fewer members. We term minimised a minimal cover with the property that substituting for a member of the cover by a proper subgroup of that member produces a collection which is no longer a cover. We here describe the minimised covers for the groups GL 2 (q), SL 2 (g), PSL 2 (<?) and PGL 2 ( 9 ).

On finite simple groups and Kneser graphs

“…Theorem 5.1 (Corollary 6.1 of [3]) When G = (P)SL(n, q) suppose that n ≥ 2 and (n, q) = (2, 5), (2,7), (2,9), (3,4). Then…”

“…Lemma 2.1 (Lemma 3.3 of [4]) Let q ≥ 4 and let G be any of the groups (P)GL (2, q), (P)SL (2, q). Then the normalizer of a Singer cycle S is the unique maximal subgroup of G containing S except when G = (P)SL (2, q) and q = 5, 7, or 9.…”

Some results and questions related to the generating graph of a finite group

“…The function σ(G) was much investigated for various non-solvable groups G (see [7], [32], [33], [20], [4]) and even for infinite groups G (see [29]). …”

The covering number of $M_{24}$

“…In [2], R. Bryce, V. Fedri, and L. Serena determined the covering numbers of some linear groups. The covering numbers of the Suzuki groups were investigated by M. S. Lucido in [9].…”