Large characteristic subgroups with modular subgroup lattice

It is known that if a group contains an abelian subgroup of finite index, then it also has an abelian characteristic subgroup of finite index. The aim of this paper is to prove that corresponding results hold when abelian subgroups are replaced either by subgroups having a modular subgroup lattice or by quasihamiltonian subgroups.     

On a class of lattice subgroup functors

In the paper, continuummany natural transitive lattice subgroup functors corresponding to no hereditary lattice formation are constructed on the class of all finite groups. This result is an answer to Question 15.39 in “The Kourovka Notebook”, which was posed by the author and A. F. Vasil’ev in connection with their theorem claiming that, on the class of all solvable groups, all functors of this kind correspond to hereditary local lattice formations.     

On subgroups containing non-trivial normal subgroups

We prove that ifA≠1 is a subgroup of a finite groupG and the order of an element in the centralizer ofA inG is strictly larger (larger or equal) than the index [G:A], thenA contains a non-trivial characteristic (normal) subgroup ofG. Consequently, ifA is a stabilizer in a transitive permutation group of degreem>1, thenexp(Z(A))<m. These theorems generalize some recent results of Isaacs and the authors.     

On the normal subgroup lattice of a hypercentral p-group

We describe all hypercentral p-groups G whose lattice of normal subgroups n(G) is isomorphic to n(H) for a group H with hypercenwal derived subgroup and H not a pgroup.     

Detecting the index of a subgroup in the subgroup lattice

A theorem by Zacher and Rips states that the finiteness of the index of a subgroup can be described in terms of purely lattice-theoretic concepts. On the other hand, it is clear that if  is a group and is a subgroup of finite index of , the index cannot be recognized in the lattice of all subgroups of , as for instance all groups of prime order have isomorphic subgroup lattices. The aim of this paper is to give a lattice-theoretic characterization of the number of prime factors (with multiplicity) of .

     

The lattice of closure operators on a subgroup lattice

We say a lattice L is a subgroup lattice if there exists a group G such that Sub(G)?L, where Sub(G) is the lattice of subgroups of G, ordered by inclusion. We prove that the lattice of closure operators which act on the subgroup lattice of a finite group G is itself a subgroup lattice if and only if G is cyclic of prime power order.     

On approximation of Lie groups by discrete subgroups

A locally compact group G is said to be approximated by discrete subgroups (in the sense of Tôyama) if there is a sequence of discrete subgroups of G that converges to G in the Chabauty topology (or equivalently, in the Vietoris topology). The notion of approximation of Lie groups by discrete subgroups was introduced by Tôyama in Kodai Math. Sem. Rep. 1 (1949) 36–37 and investigated in detail by Kuranishi in Nagoya Math. J. 2 (1951) 63–71. It is known as a theorem of Tôyama that any connected Lie group approximated by discrete subgroups is nilpotent. The converse, in general, does not hold. For example, a connected simply connected nilpotent Lie group is approximated by discrete subgroups if and only if G has a rational structure. On the other hand, if Γ is a discrete uniform subgroup of a connected, simply connected nilpotent Lie group G then G is approximated by discrete subgroups Γ _n containing Γ. The proof of the above result is by induction on the dimension of G, and gives an algorithm for inductively determining Γ _n . The purpose of this paper is to give another proof in which we present an explicit formula for the sequence (Γ _n ) _n?≥?0 in terms of Γ. Several applications are given.     

On the set of discrete subgroups of bounded covolume in a semisimple group

In this noteG is a locally compact group which is the product of finitely many groups G_s(k_s)(s∈S), where k_s is a local field of characteristic zero and G_s an absolutely almost simplek _s-group, ofk _s-rank ≥1. We assume that the sum of the r_s is ≥2 and fix a Haar measure onG. Then, given a constantc > 0, it is shown that, up to conjugacy,G contains only finitely many irreducible discrete subgroupsL of covolume ≥c (4.2). This generalizes a theorem of H C Wang for real groups. His argument extends to the present case, once it is shown thatL is finitely presented (2.4) and locally rigid (3.2).     

Solvable subgroups in groups with self-normalizing subgroup

We study the structure of some solvable finite subgroups in groups with self-normalizing subgroup. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1716–1721, December, 2008.