Subgroup coverings of some sporadic groups

Abstract

A set of proper subgroups is a covering for a group if their union is the whole group. Determining the size of a smallest covering is an open problem for many simple groups. For some of the sporadic groups, we find subgroup coverings of minimal cardinality. For others we specify the isomorphism types of subgroups in a smallest covering and use graphs to calculate bounds for its size.

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]). …”

“…Theorems 9.2 and 9.3 can be used to consider Problem B in various situations, for example to find ω(G) for certain sporadic simple groups G. We note that the exact value of σ(M 23 ) is computed in [20]. Now let k > 2.…”

Some Remarks on Groups With Nilpotent Minimal Covers

“…https://doi.org/10.1017/S1446788708000670 [9] Some remarks on groups with nilpotent minimal covers 361…”

“…(24) The O'Nan-Sims group G = O'N. Here |G| = 2 9 · 3 4 · 5 · 7 3 · 11 · 19 · 31 and N G (C 11 ) has order 110 (see [17, p. 225]) and it is not maximal (see [8, p. 132] 14 · 3 6 · 5 6 · 7 · 11 · 19 and C G (C 7 ) = C 7 × Alt 5 is not nilpotent (see [17, p. 226]). …”

The covering number of $M_{24}$

“…In [6], P. E. Holmes determined the covering numbers of the Mathieu groups M 11 , M 22 , and M 23 , as well as the Lyons group and the O'Nan group, and gave upper and lower bounds for the covering numbers of the Janko group J 1 and the McLaughlin group. The covering number of M 12 was determined by L. C. Kappe, D. Nikolova-Popova, and E. Swartz in [8].…”