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Let FL_s(K) be the finitary linear group of degree s over an associative ring K with unity. We prove that the torsion subgroups of FL_s(K) are locally finite for certain classes of rings K. A description of some f.g. solvable subgroups of FL_s(K) are given.
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We count the finitely generated subgroups of the modular group $textsf{PSL}(2,mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $Gamma(H)$, we consider the number of vertices of $Gamma(H)$ to be the size of $H$ and we count the subgroups of size $n$. Since an index $n$ subgroup has size $n$, our results generalize the known results on the enumeration of the finite index subgroups of $textsf{PSL}(2,mathbb{Z})$. We give asymptotic equivalents for the number of finitely generated subgroups of $textsf{PSL}(2,mathbb{Z})$, as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size $n$ subgroup and prove a large deviations statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size $n$ subgroup (resp. finite index subgroup, free subgroup) of $textsf{PSL}(2,mathbb{Z})$.
We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $cap H_i={1}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.
We show that finitely-generated, purely pseudo-Anosov subgroups of the genus-2 Goeritz group are convex cocompact in the genus-2 mapping class group.
Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $delta_k$-values $a,bin G$ of coprime orders. In the course of the proof we establish the following result of independent interest: Let $P$ be a Sylow $p$-subgroup of $G$. Then $Pcap G^{(k)}$ is generated by $delta_k$-values contained in $P$. This is related to the so-called Focal Subgroup Theorem.
This paper is a new contribution to the study of regular subgroups of the affine group $AGL_n(F)$, for any field $F$. In particular we associate to any partition $lambda eq (1^{n+1})$ of $n+1$ abelian regular subgroups in such a way that different partitions define non-conjugate subgroups. Moreover, we classify the regular subgroups of certain natural types for $nleq 4$. Our classification is equivalent to the classification of split local algebras of dimension $n+1$ over $F$. Our methods, based on classical results of linear algebra, are computer free.
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