In this short note, it is shown that if $A,B$ are $H$-connected transversals for a finite subgroup $H$ of an infinite group $G$ such that the index of $H$ in $G$ is at least 3 and $H\cap H^u\cap H^v=1$ whenever $u,v\in G\setminus H$ and $u{v}^{-1}\in G\setminus H$ then $A=B$ is a normal abelian subgroup of $G$.

Subnormal subgroups possessing connected transversals are briefly discussed.

In this article, conservation rules of the direct sum decomposition of groups are mainly discussed. In the first section, we prepare miscellaneous definitions and theorems for further formalization in Mizar [5]. In the next three sections, we formalized the fact that the property of direct sum decomposition is preserved against the substitutions of the subscript set, flattening of direct sum, and layering of direct sum, respectively. We referred to [14], [13] [6] and [11] in the formalization.

In this paper we study an action $D$ of the absolute Galois group $\Gamma =Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action $D$ is induced by the action of $\Gamma$ on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action $D$ and compare it with the Grothendieck action.

Étant donné un automorphisme $\sigma$ d’un groupe libre et un représentant topologique train-track de son inverse, on peut construire un arbre réel $T$ appelé arbre répulsif de $\sigma$. Le groupe libre agit sur $T$ par isométries. La dynamique engendrée par $\sigma$ peut être représentée par l’action du groupe libre restreinte à un sous-ensemble compact bien choisi du complété métrique de $T$. Cet article construit ce sous-ensemble sur une classe d’exemples en introduisant des opérations appelées substitutions d’arbre ;...

For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian quotient, called the Betti number. We show that any combination of the group's rank, co-rank, and Betti number within obvious constraints is realized for some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number is additive...

Let $K$ be a $p$-adic field, and let $H=PS{L}_2\left(K\right)$ endowed with the Haar measure determined by giving a maximal compact subgroup measure $1$. Let $A{L}_H\left(x\right)$ denote the number of conjugacy classes of arithmetic lattices in $H$ with co-volume bounded by $x$. We show that under the assumption that $K$ does not contain the element $\zeta +{\zeta }^{-1}$, where $\zeta$ denotes the $p$-th root of unity over ${\mathbb{Q}}_p$, we have $$\underset{x\to \infty }{lim}\frac{logA{L}_H\left(x\right)}{xlogx}=q-1$$ where $q$ denotes the order of the residue field of $K$.

We introduce the notion of a critical constant $c_{rt}$ for recurrence of random walks on $G$-spaces. For a subgroup $H$ of a finitely generated group $G$ the critical constant is an asymptotic invariant of the quotient $G$-space $G/H$. We show that for any infinite $G$-space $c_{rt}\ge 1/2$. We say that $G/H$ is very small if $c_{rt}\<1$. For a normal subgroup $H$ the quotient space $G/H$ is very small if and only if it is finite. However, we give examples of infinite very small $G$-spaces. We show also that critical constants for recurrence can be used...

All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given.