相关论文: On a conjecture of Karrass and Solitar

In this note we give a particularly short and simple proof of the following theorem of Karrass and Solitar. Let $H$ be a finitely generated subgroup of a free group $F$ with infinite index $[F:H]$. Then there is a nontrivial normal subgroup…

We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F, there is a subgroup H of finite index in B such that the subgroup generated by A and H has infinite index in F. The main corollary…

We prove that the reduced Kurosh rank of the intersection of two subgroups $H$ and $K$ of a free product of right-orderable groups is bounded above by the product of the reduced Kurosh ranks of $H$ and $K$. In particular, taking the…

We show that given a finitely generated LERF group $G$ with positive rank gradient, and finitely generated subgroups $A,B \leq G$ of infinite index, one can find a finite index subgroup $B_0$ of $B$ such that $[G : \langle A \cup B_0…

A subgroup $H$ of a group $G$ is said to be {pronormal} in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. Some problems in finite group theory, combinatorics, and permutation group theory were solved in…

A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend…

The following results are proved: The center of any finite index subgroup of an irreducible, infinite, non-affine Coxeter group is trivial; Any finite index subgroup of an irreducible, infinite, non-affine Coxeter group cannot be expressed…

Problem 20.21 of Mazurov and Khukhro (Unsolved Problems in Group Theory: The Kourovka Notebook, 20th Issue, 2022), contributed by M.~Conder and attributed to G.~Verret, asks whether there exists a finite group $G$ with two normal subgroups…

A generalized Baumslag-Solitar (GBS) group is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. We show how to determine effectively the rank (minimal cardinality of a generating set) of a GBS…

The object of this note is to give a very short proof of the following theorem of Ivanov and Schupp. Let H be a finitely generated subgroup of a free group F and the index [F:H] infinite. Then there exists a nontrivial normal subgroup N of…

We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank $n$ group $G$ and $H$ has index at least $n$ in $G$ then we can construct a left transversal for $H$ which…

In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\leq G$ with index $[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which…

Given nontrivial finite groups $A$ and $B$, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product $A \ast B$. To show this, we prove that every finite simple group of…

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of…

We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups…

We carry out a study of groups $G$ in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite…

The goal of this note is to provide yet another proof of the following theorem of Golod: there exists an infinite finitely generated group $G$ such that every element of $G$ has finite order. Our proof is based on the Nielsen-Schreier index…

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is…

A group $G$ is invariably generated (IG) if there is a subset $S \subseteq G$ such that for every subset $S' \subseteq G$, obtained from $S$ by replacing each element with a conjugate, $S'$ generates $G$. $G$ is finitely invariably…

Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If…