Groups whose all subgroups are ascendant or self-normalizing

This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16].     

On the Pronormality of Subgroups of Odd Index in Some Extensions of Finite Groups

We study finite groups with the following property (*): All subgroups of odd index are pronormal. Suppose that G has a normal subgroup A with property (*), and the Sylow 2-subgroups of G/A are self-normalizing. We prove that G has property (*) if and only if so does N_G(T)/T, where T is a Sylow 2-subgroup of A. This leads to a few results that can be used for the classification of finite simple groups with property (*).     

Abnormal,Pronormal, Contranormal,and Carter Subgroups in Some Generalized Minimax Groups

ABSTRACT

Some properties of abnormal and pronormal subgroups in generalized minimax groups are considered. For generalized minimax groups (not only periodic) whose locally nilpotent residual is nilpotent and satisfies Min-G the existence of Carter subgroups and their conjugations have been proven. Some generalizations of results of J. Rose on abnormal and contranormal subgroups have been also obtained.     

On some groups all subgroups of which are nearly pronormal

A subgroup H of a group G is called nearly pronormal in G if, for every subgroup L of the group G that contains H, the normalizer N _L (H) is contranormal in L. We prove that if G is a (generalized) solvable group in which every subgroup is nearly pronormal, then all subgroups of G are pronormal. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1331–1338, October, 2007.     

On weakly S-embedded and weakly τ-embedded subgroups

Let G be a finite group. A subgroup H of G is said to be weakly S-embedded in G if there exists a normal subgroup K of G such that HK is S-quasinormal in G and H ∩ K ≤ H _seG , where H _seG is the subgroup generated by all those subgroups of H which are S-quasinormally embedded in G. We say that a subgroup H of G is weakly τ-embedded in G if there exists a normal subgroup K of G such that HK is S-quasinormal in G and H ∩ K ≤ H _seG , where H _seG is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. In this paper, we study the properties of weakly S-embedded and weakly τ-embedded subgroups, and use them to determine the structure of finite groups.     

On self-normality and abnormality in the alternating group Ap

An old problem proposed by Huppert, Doerk and Hawkes motivates us to investigate the relationship between an abnormal subgroup and self-normalizing in non-solvable groups. A subgroup H of a group G is called second maximal if H is maximal in all maximal subgroups of G containing H. Our result is that if H is a second maximal subgroup of the alternating group A_p of prime degree, then H is abnormal in A_p if and only if H is self-normalizing.     

Groups with every subgroup ascendant-by-finite

A subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.     

Joint spectrum and the infinite dihedral group

A subgroup H of a group G is called pronormal if, for any element g ∈ G, the subgroups H and H ^g are conjugate in the subgroup <H,H ^g >. We prove that, if a group G has a normal abelian subgroup V and a subgroup H such that G = HV, then H is pronormal in G if and only if U = N _U (H)[H,U] for any H-invariant subgroup U of V. Using this fact, we prove that the simple symplectic group PSp_6n (q) with q ≡ ±3 (mod 8) contains a nonpronormal subgroup of odd index. Hence, we disprove the conjecture on the pronormality of subgroups of odd indices in finite simple groups, which was formulated in 2012 by E.P. Vdovin and D.O. Revin and verified by the authors in 2015 for many families of simple finite groups.     

Connectivity of intersection graphs of finite groups

The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if H∩K≠1 where 1 denotes the trivial subgroup of G. In this paper, we classify finite solvable groups whose intersection graphs are not 2-connected and finite nilpotent groups whose intersection graphs are not 3-connected. Our methods are elementary.     

Structure of locally graded CDN (]-groups

We introduce the notion of a CDN(]-group G, namely, a group such that, for any pair of its subgroups A and B such that A is a proper nonmaximal subgroup of B, there exists a normal subgroup N of G and A < N ≤ B. Thirteen types of non-Dedekind nilpotent groups and 9 types of nonnilpotent locally graded groups of this kind are described. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1532–1536, November, 1998.     

Products of F*(<Emphasis Type="Italic">G</Emphasis>)-subnormal subgroups of finite groups

A subgroup H of a finite group G is called F*(G)-subnormal if H is subnormal in HF*(G). We show that if a group Gis a product of two F*(G)-subnormal quasinilpotent subgroups, then G is quasinilpotent. We also study groups G = AB, where A is a nilpotent F*(G)-subnormal subgroup and B is a F*(G)-subnormal supersoluble subgroup. Particularly, we show that such groups G are soluble.     

The Subgroup Normalizer Problem for Integral Group Rings of Some Nilpotent and Metacyclic Groups

For a group G and a subgroup H of G, this article discusses the normalizer of H in the units of a group ring RG. We prove that H is only normalized by the “obvious” units, namely products of elements of G normalizing H and units of RG centralizing H, provided H is cyclic. Moreover, we show that the normalizers of all subgroups of certain nilpotent and metacyclic groups in the corresponding group rings are as small as possible. These classes contain all dihedral groups, all finite nilpotent groups, and all finite groups with all Sylow subgroups being cyclic.     

On Nearly SS-Embedded Subgroups of Finite Groups

Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H ∩ K ≤ HseG, where HseG is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly SS-embedded subgroups on the structure of finite groups.     

On the restriction map of the Fourier-Stieltjes algebra B(G) and Bp(G)

G is a locally compact group that contains the semidirect product J of a closed normal subgroup H and a closed connected subgroup K. Conditions on J are given that imply that the restriction map B_p(G) → B_p(H) (1 < p < ∞; G amenable if p ≠ 2) of the Fourier-Stieltjes algebras is not surjective. It is also shown that if the restriction map B(J) → B(H) is surjective, J need not be a direct product, even if H is nilpotent.     

On SS-quasinormal and S-quasinormally embedded subgroups of finite groups

A subgroup H of a group G is said to be an SS-quasinormal (Supplement-Sylow-quasinormal) subgroup if there is a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B. A subgroup H of a group G is said to be S-quasinormally embedded inGif for every Sylow subgroup P of H, there is an S-quasinormal subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain SS-quasinormal or S-quasinormally embedded subgroups of prime power order are studied.     

Finite <Emphasis Type="Italic">p</Emphasis>-groups with a class of complemented normal subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.     

A Note on Solitary Subgroups of Finite Groups

We say that a subgroup H of a finite group G is solitary (respectively, normal solitary) when it is a subgroup (respectively, normal subgroup) of G such that no other subgroup (respectively, normal subgroup) of G is isomorphic to H. A normal subgroup N of a group G is said to be quotient solitary when no other normal subgroup K of G gives a quotient isomorphic to G/N. We show some new results about lattice properties of these subgroups and their relation with classes of groups and present examples showing a negative answer to some questions about these subgroups.     

Ideals with bounded approximate identities in Fourier algebras

We make use of the operator space structure of the Fourier algebra A(G) of an amenable locally compact group to prove that if H is any closed subgroup of G, then the ideal I(H) consisting of all functions in A(G) vanishing on H has a bounded approximate identity. This result allows us to completely characterize the ideals of A(G) with bounded approximate identities. We also show that for several classes of locally compact groups, including all nilpotent groups, I(H) has an approximate identity with norm bounded by 2, the best possible norm bound.