Si determinano i gruppi finiti il cui insieme parzialmente ordinato delle classi di coniugio dei sottogruppi è isomorfo a quello di un gruppo abeliano.
Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups whose set of numbers of subgroups of possible orders has exactly two elements. We show that if is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then has a normal Sylow subgroup of prime order and is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with...
In this paper, we consider finite groups with precisely one nonlinear nonfaithful irreducible character. We show that the groups of order 16 with nilpotency class 3 are the only -groups with this property. Moreover we completely characterize the nilpotent groups with this property. Also we show that if is a group with a nontrivial center which possesses precisely one nonlinear nonfaithful irreducible character then is solvable.
In this paper it is proved that a finite group G with an automorphism of prime order r, such that is contained in a nilpotent subgroup H, with , is nilpotent provided that either is odd or, if is even, then r is not a Fermât prime.
This Note contains the complete list of finite groups, having exactly eight non-linear irreducible characters. In section 4 we consider in full details some typical cases.
We describe finite groups which contain just one conjugate class of self-normalizing subgroups.
The notions of permutable and globally permutable lattices were first introduced and studied by J. Krempa and B. Terlikowska-Osłowska [4]. These are lattices preserving many interesting properties of modular lattices. In this paper all finite groups with globally permutable lattices of subgroups are described. It is shown that such finite p-groups are exactly the p-groups with modular lattices of subgroups, and that the non-nilpotent groups form an essentially larger class though they have a description...
In 1954, Kontorovich and Plotkin introduced the concept of a modular chain in a lattice to obtain a lattice-theoretic characterization of the class of torsion-free nilpotent groups. We determine the structure of finite groups with modular chains. It turns out that this class of groups lies strictly between the class of finite groups with lower semimodular subgroup lattice and the projective closure of the class of finite nilpotent groups.
In this paper we consider a prime graph of finite groups. In particular, we expect finite groups with prime graphs of maximal diameter.
We prove that are primitive the finite groups whose normalizers of the Sylow subgroups are primitive. We classify the groups of such class, denoted by , and we study the Schunck classes whose boundary is contained in . We give also necessary and sufficient conditions in order that the projectors be subnormally embedded.
Let be a finite group, the smallest prime dividing the order of and a Sylow -subgroup of with the smallest generator number . There is a set of maximal subgroups of such that . In the present paper, we investigate the structure of a finite group under the assumption that every member of is either -permutably embedded or weakly -permutable in to give criteria for a group to be -supersolvable or -nilpotent.
A subgroup of a finite group is said to be SS-supplemented in if there exists a subgroup of such that and is S-quasinormal in . We analyze how certain properties of SS-supplemented subgroups influence the structure of finite groups. Our results improve and generalize several recent results.