On the subgroup index problem in finite groups

We address the subgroup index problem in a given finite subgroup lattice L = ℓ(G) which is P-indecomposable and determine out of the structure of L the existence in G of a subgroup [(D)tilde]tilde D invariant for all automorphisms of L, with a cyclic complement R in G and where for any pair X ≤ Y of subgroups of [(D)tilde]tilde D the index |Y: X| can be computed using only structural properties of L. As a consequence, we show that in such an L all the terms of the Fitting series of G can be determined, as well as an upper bound of the order of G can be computed out of L as long as G has no cyclic Hall direct factor.     

On subgroup functors of finite soluble groups

The principal aim of this paper is to study the regular and transitive subgroup functors in the universe of all finite soluble groups. We prove that they form a complemented and non-modular lattice containing two relevant sublattices. This is the answer to a question (Question 1.2.12) proposed by Skiba (1997) in the finite soluble universe.     

On intervals in subgroup lattices of finite groups

We investigate the question of which finite lattices are isomorphic to the lattice of all overgroups of a subgroup in a finite group . We show that the structure of is highly restricted if is disconnected. We define the notion of a ``signalizer lattice" in and show for suitable disconnected lattices , if is minimal subject to being isomorphic to or its dual, then either is almost simple or admits a signalizer lattice isomorphic to or its dual. We use this theory to answer a question in functional analysis raised by Watatani.

     

On the congruence subgroup problem for branch groups

We prove here an energy estimate for the Cauchy problem for hyperbolic equations with double characteristic, which contains both effectively and non-effectively points (see L. Hörmander [3] and R. Melrose [8]) in a unique framework.     

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 .

     

On the lattice automorphisms of the finite Chevalley groups

A lattice automorphism of a group is defined to be an automorphism of its lattice of subgroups. For a large class of finite simple Chevalley groups, it is shown that every lattice automorphism is induced by a group automorphism. However, this does not hold for all finite simple Chevalley groups G, as is shown by explicit construction in the case G=PSL(3, q).     

On finite groups whose derived subgroup has bounded rank

Let G be a finite group with derived subgroup of rank r. We prove that |G: Z ₂(G)| ≤ |G′|^2r . Motivated by the results of I. M. Isaacs in [5] we show that if G is capable then |G: Z(G)| ≤ |G′|^4r . This answers a question of L. Pyber. We prove that if G is a capable p-group then the rank of G/Z(G) is bounded above in terms of the rank of G′.     

On the existence of a free subgroup of finite index

Let G be a finitely generated accessible group. We will study the homology of G with coefficients in the left G-module H¹(G;$\text{Z}$[G]). This G-module may be identified with the G-module of continuous functions with values in $\text{Z}$ on the G-space of ends of G, quotiented by the constant functions. The main result is as follows: Suppose G is infinite, then the abelian group H₁(G;H¹(G;$\text{Z}$[G])) has rank 1 if G has a free subgroup of finite index and it has rank 0 if G has not.     

On solvability of groups with a finite nilpotent supercomplemented subgroup

We investigate groups in which every subgroup containing some fixed finite nilpotent subgroup has a complement.     

On the number of edges of cyclic subgroup graphs of finite groups

Archiv der Mathematik - In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of...     

On the lattice of all totally composition formations of finite groups

It is shown that the lattice of all totally composition formations of finite groups is algebraic.

     

On the shellability of the order complex of the subgroup lattice of a finite group

We show that the order complex of the subgroup lattice of a finite group is nonpure shellable if and only if is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups.

     

On subgroups of finite index in compact Hausdorff groups

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.