On groups with an irredundant 7-cover

Abstract

A cover for a group is a collection of proper subgroups whose union is the whole group. A cover is irredundant if no proper sub-collection is also a cover and is called maximal if all its members are maximal subgroups. For an integer n > 2, a cover with n members is called an n-cover. In this paper we determine groups with a maximal irredundant 7-cover with core-free intersection. The intersection of an irredundant n-cover is known to have index bounded by a function of n, though in general the precise bound is not known. Here we prove that the exact bound is 81 when n is 7.

B.H. Neumann's Question on Ensuring Commutativity of Finite Groups

“…If G = PSL(2,p) and p > 7 (p is a prime number), then[1, Lemma 4.4] implies that w(G) ^ 133, a contradiction. If G = PSL(2,7), then by [1, Proposition 3.21] and a similar argument as for A 5 we conclude that G is not a C(m, n)-group. If G = Sz(2 p ), then \G\ = 2 2 * x (V -1) x (2 2p -1-1) and u 2 (G) = 2 2p + 1 ^ 65 (see Theorem 3.10 (and its proof) of[8, Chapter XI]).…”

“…Then G = C G {ai) U C G (a 2 ) U C G (a 3 ) U Cofa) U C G {a 5 ) U C G (a 6 ) U C G (b). Now by [2,Theorem B], \G\ ^ 81. But |G| = 2* • 3 and hence |G| 6 {6,12,24,48}.…”

On 10-Centralizer Groups of Odd Order

“…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.…”

Identification some groups with special subgroups

“…Then G has a maximal irredundant 6cover with core-free intersection D if and only if G satisfies one of the following properties. Abdollahi et al [3], characterized groups with maximal irredundant 7-cover with core-free intersection.…”