ON MINIMAL WEAKLY s-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS

Suppose G is a finite group and H is subgroup of G. H is said to be s-permutable in G if HGp = GpH for any Sylow p-subgroup Gp of G; H is called weakly s-supplemented subgroup of G if there is a subgroup T of G such that G = HT and H ∩ T ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of minimal weakly s-supplemented subgroups on the structure of finite groups and generalize some recent results. Furthermore, we give a positive answer in the minimal subgroup case for Skiba's Open Questions in [On weakly s-permutable subgroups of finite groups, J. Algebra315 (2007) 192–209].

ON MINIMAL SUBGROUP COVERINGS OF SOME AFFINE PRIMITIVE PERMUTATION GROUPS

A cover for a group G is a collection of proper subgroups of G whose union is G. Cohn defined σ(G) to be the least integer m such that G is the union of m proper subgroups. Determining σ is an open problem for most non-solvable groups. We obtain σ(G) for some affine primitive groups.

ON MINIMAL SUBGROUPS OF FINITE GROUPS

Let G be a finite group. A minimal subgroup of G is a subgroup of prime order. A subgroup of G is called S-quasinormal in G if it permutes with each Sylow subgroup of G. A group G is called an MS-group if each minimal subgroup of G is S-quasinormal in G. In this paper, we investigate the structure of minimal non-MS-groups (non-MS-groups all of whose proper subgroups are MS-groups).

Further Duals of a verbal Subgroup

In a previous paper [3] we gave two methods for constructing subgroups which in certain senses may be considered to be dual to a verbal subgroup Vf(G) of an arbitrary group G. Associated with a word h (u, v) in the two symbols u and v, we have (i) the first dual subgroup which is defined as the minimal subgroup of G containing all elements ξ of G for whichfor all values of x1, x2, …, in xn in G, and (ii), the second dual subgroup which is defined as the minimal subgroup of G containing all elements z of G for whichfor all values of x1, x2, …, xn in G. Below we introduce slight variations to these definitions, which give rise to the concepts of the third and the fourth dual subgroups respectively. For certain values of h(u, v) we obtain concepts which also arise from and , namely, the marginal subgroup, the invariable subgroup and the centralizer of a verbal subgroup. We also obtain the new concepts of elemental subgroups and commutal subgroups and briefly sketch some of their properties. Finally we conclude by showing that MacLane's dual for the centralizer of a verbal subgroup is a closely related verbal subgroup.