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.     

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 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.     

On the subgroup distance problem

We investigate the computational complexity of finding an element of a permutation group H⊆S_n with minimal distance to a given π∈S_n, for different metrics on S_n. We assume that H is given by a set of generators. In particular, the size of H might be exponential in the input size, so that in general the problem cannot be solved in polynomial time by exhaustive enumeration. For the case of the Cayley Distance, this problem has been shown to be NP-hard, even if H is abelian of exponent two [R.G.E. Pinch, The distance of a permutation from a subgroup of S_n, in: G. Brightwell, I. Leader, A. Scott, A. Thomason (Eds.), Combinatorics and Probability, Cambridge University Press, 2007, pp. 473-479]. We present a much simpler proof for this result, which also works for the Hamming Distance, the l_p distance, Lee’s Distance, Kendall’s tau, and Ulam’s Distance. Moreover, we give an NP-hardness proof for the l_∞ distance using a different reduction idea. Finally, we discuss the complexity of the corresponding fixed-parameter and maximization problems.     

On the membership problem of finite groups in solvably saturated formations

Theorems on the inclusion of a finite group G in a solvably saturated formation $\mathfrak{F}$ are proved in terms of $\mathfrak{F}$ -centrality and c-normality.     

The congruence subgroup problem for branch groups

We state and study the congruence subgroup problem for groups acting on rooted trees, and for branch groups in particular. The problem is reduced to the computation of the congruence kernel, which we split into two parts: the branch kernel and the rigid kernel. In the case of regular branch groups, we prove that the first one is abelian while the second has finite exponent. We also establish some rigidity results concerning these kernels.