A subgroup of a group is said to be complemented in if there exists a subgroup of such that and . In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about -nilpotent groups.
Let be a finite group. A normal subgroup of is a union of several -conjugacy classes, and it is called -decomposable in if it is a union of distinct -conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its...
Let be a finite group with a dicyclic subgroup . We show that if there exist -connected transversals in , then is a solvable group. We apply this result to loop theory and show that if the inner mapping group of a finite loop is dicyclic, then is a solvable loop. We also discuss a more general solvability criterion in the case where is a certain type of a direct product.
We show that finite commutative inverse property loops may not have nonabelian dihedral 2-groups as their inner mapping group.
The major aim of the present paper is to strengthen a nice result of Shemetkov and Skiba which gives some conditions under which every non-Frattini G-chief factor of a normal subgroup E of a finite group G is cyclic. As applications, some recent known results are generalized and unified.
Let F be a Galois extension of a number field k with the Galois group G. The Brauer-Kuroda theorem gives an expression of the Dedekind zeta function of the field F as a product of zeta functions of some of its subfields containing k, provided the group G is not exceptional. In this paper, we investigate the exceptional groups. In particular, we determine all nilpotent exceptional groups, and give a sufficient condition for a group to be exceptional. We give many examples of nonnilpotent solvable...