We count the finitely generated subgroups of modular group [Formula: see text]. More precisely, each such subgroup text] can be represented by its Stallings graph text], we consider number vertices to size and Since an index has our results generalize known on enumeration finite give asymptotic equivalents for as well subgroups, free subgroups. also expected value isomorphism type a prove large...

A filling subgroup of a finitely generated free group F (X) is a subgroup which does not fix a point in any very small action free action on an R-tree. For the free group of rank two, we construct a combinatorial algorithm to determine whether or not a given finitely generated subgroup is filling. In higher ranks, we discuss two types of non-filling subgroups: those contained in loop vertex sub...

Let Γ be a finitely generated linear group over a field of characteristic 0. Suppose that every solvable subgroup of Γ is polycyclic. Then any solvable subgroup of Γ is separable. This conclusion is false without the hypothesis that every solvable subgroup of Γ is polycyclic.

Motivated by cryptography, permutations induced by polynomial functions on finite lattices L = (L,∨,∧,∗ ) with an antitone involution ∗ are investigated. These permutations together with the operation of composition form a subgroup of the symmetric group on L. We describe the structure of this subgroup for different classes of lattices L and indicate possible applications by outlining a protoco...

let $h$, $l$ and $x$ be subgroups of a finite group$g$. then $h$ is said to be $x$-permutable with $l$ if for some$xin x$ we have $al^{x}=l^{x}a$. we say that $h$ is emph{$x$-quasipermutable } (emph{$x_{s}$-quasipermutable}, respectively) in $g$ provided $g$ has a subgroup$b$ such that $g=n_{g}(h)b$ and $h$ $x$-permutes with $b$ and with all subgroups (with all sylowsubgroups, respectively) $v$...

let  $varpi$ be a representation of the homogeneous space $g/h$, where $g$ be a locally compact group and  $h$ be a compact subgroup of $g$. for  an admissible wavelet $zeta$ for $varpi$  and $psi in l^p(g/h), 1leq p

In this paper, the reflection of ultrasonic guided waves from simple dent in pipes has been investigated using finite element method and the relationship between reflection coefficient of these waves and deformation rate has been determined. Also, the effect of the parameters of wave generation source on the generated wave field has been investigated using normal modes expansion method. At firs...

Let $G=\operatorname {SO}^\circ (n,1) \times \operatorname (n,1)$ and $X=\mathbb {H}^{n}\times \mathbb {H}^{n}$ for $n\ge 2$. For a pair $(\pi _1, \pi _2)$ of non-elementary convex cocompact representations finitely generated group $\Sigma$ into $\operatorname (n,1)$, let $\Gamma =(\pi _1\times _2)(\Sigma )$. Denoting the bottom $L^2$-spectrum negative Laplacian on {\setminus } X$ by $\lambda _...

For many groups the structure of finitely generated subgroups is generically simple. That is with asymptotic density equal to one a randomly chosen finitely generated subgroup has a particular well-known and easily analyzed structure. For example a result of D. B. A. Epstein says that a finitely generated subgroup of GL(n,R) is generically a free group. We say that a group G has the generic fre...

We construct explicitly APF extensions of finite extensions of Qp for which the Galois group is not a p-adic Lie group and which do not have any open subgroup with Zp-quotient. Let K be a finite extension of Qp and L/K be a Galois totally ramified pro-pextension. If the Galois group of L/K is a p-adic Lie group it is known from Sen ([Sen]) that the sequence of upper ramification breaks of L/K i...