Covering groups with subgroups

Abstract

A group is covered by a collection of subgroups if it is the union of the collection. The intersection of an irredundant cover of n subgroups is known to have index bounded by a function of n, though in general the precise bound is not known. Here we confirm a claim of Tompkinson that the correct bound is 16 when n is 5. The proof depends on determining all the 'minimal' groups with an irredundant cover of five maximal subgroups.

On 10-Centralizer Groups of Odd Order

“…For a natural number , let ( ) denote the largest index | : |, where is a group with an irredundant -cover whose intersection of all of them is . We know that (3) = 4, (4) = 9, (5) = 16, and (6) = 36 (see [12][13][14][15], resp.). Now we present some lemmas and propositions that will be used in the proof of Theorem 1.…”

“…Now we present some lemmas and propositions that will be used in the proof of Theorem 1. 81 (see [12][13][14][15][16], resp. ); (4) let be a group such that every proper centralizer in is abelian.…”

On groups with an irredundant 7-cover

“…The proof of Theorem A is similar to that of the characterization of C 5 and C 6 -groups given in [2] and [1], respectively. To characterize C 7 -groups, we distinguish between three cases: nilpotent, semisimple and non-semisimple groups, where by a semisimple group we mean a group having no non-trivial normal abelian subgroup.…”

“…Bryce et al [2] and Abdollahi et al [1] characterized C n -groups for n = 5, 6 respectively. Here we characterize C 7 -groups.…”

“…(7) G ∼ = (C 2 ) 6 Sym 3 and |D| = 6. Neumann [7] proved that if G has an irredundant n-cover then the index of the intersection of the cover in G is bounded by a function of n. Tomkinson [10] improved that bound and gave estimates for f (n), the largest index |G : D| over all groups G having an irredundant n-cover with intersection D. He suggested that the lower bound In [8,6,2], the value of f (n) was obtained for n = 3, 4, 5, namely f (3) = g(3), f (4) = g(4) and f (5) = g(5), respectively. Also Abdollahi et al [1] have shown that f (6) = g (6).…”