On the probability of generating free prosoluble groups of small rank (original) (raw)

On the probability of generating a minimal d-generated group

Journal of the Australian Mathematical Society, 2001

We consider finite groups with the property that any proper factor can be generated by a smaller number of elements than the group itself. We study some problems related with the probability of generating these groups with a given number of elements.

Random Generation of Finitely Generated Subgroups of a Free Group

International Journal of Algebra and Computation, 2008

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity [Formula: see text] in the RAM model and [Formula: see text] in the bitcost model.

On the number of generators needed for free profinite products of finite groups

Israel Journal of Mathematics, 2007

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C 1 ,. .. , C n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C 1 ,. .. , C n and the minimal number of generators of G is n.

1 Growth in Free Groups (And Other Stories)–Twelve Years Later

2016

We start by studying the distribution of (cyclically reduced) elements of the free groups F n with respect to their abelianization (or equivalently, their class in H 1 (F n , Z)). We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions mod p (p-an arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.

On two distributions of subgroups of free groups

2010 Proceedings of the Seventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO), 2010

We study and compare two natural distributions of finitely generated subgroups of free groups. One is based on the random generation of tuples of reduced words; that is the one classically used by group theorists. The other relies on Stallings' graphical representation of subgroups and in spite of its naturality, it was only recently considered. The combinatorial structures underlying both distributions are studied in this paper with methods of analytic combinatorics. We use these methods to point out the differences between these distributions. It is particularly interesting that certain important properties of subgroups that are generic in one distribution, turn out to be negligible in the other.

Homogeneous number of free generators

Journal of Group Theory, 2000

We address two questions of Simon Thomas. First, we show that for any n ≥ 3 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group Γ which admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group Γ is uncountable.

Growth in free groups (and other stories)—twelve years later

Illinois Journal of Mathematics, 2010

We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their Abelianization (or equivalently, their class in H1(Fn, Z)). We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions mod p (pan arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.