Related papers: Groups with Tarski number 5

An element $g$ of a finite group $G$ is said to be vanishing in $G$ if there exists an irreducible character $\chi$ of $G$ such that $\chi(g)=0$; in this case, $g$ is also called a zero of $G$. The aim of this paper is to obtain structural…

Let $G$ be a finite group and $A$ be a normal subgroup of $G$. We denote by $ncc(A)$ the number of $G$-conjugacy classes of $A$ and $A$ is called $n$-decomposable, if $ncc(A)=n$. Set ${\cal K}_G = \{ncc(A)| A \lhd G \}$. Let $X$ be a…

A $c$-labeling $\phi: V(G) \rightarrow \{1, 2, \hdots, c \}$ of graph $G$ is distinguishing if, for every non-trivial automorphism $\pi$ of $G$, there is some vertex $v$ so that $\phi(v) \neq \phi(\pi(v))$. The distinguishing number of $G$,…

We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)| is at most f(k), where Sol(G) denotes the largest solvable normal…

The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…

The variety $JorN_{5}$ of five-dimensional nilpotent Jordan algebras structures over an algebraically closed field is investigated. We show that $JorN_{5}$ is the union of five irreducible components, four of them correspond to the Zariski…

The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the least integer $n$ such that $G$ is isomorphic to a subgroup of the symmetric group $S_n$. If $G$ has a normal subgroup $N$ such that $\mu(G/N) > \mu(G)$, then $G$…

We produce a simple group $G$ of cardinality $\aleph_1$ which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset $Y \subseteq G$ there exists a natural…

There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, the average number of zeros of characters of a finite group $ G $ as the number of zeros…

We call a group $G$ {\it algorithmically finite} if no algorithm can produce an infinite set of pairwise distinct elements of $G$. We construct examples of recursively presented infinite algorithmically finite groups and study their…

For a group G and positive interger m, Gm denotes the subgroup generated by the elements gm where g runs through G. The subgroups not of the form Gm are called nonpower subgroups. We extend the classification of groups with few nonpower…

The function $\mathrm{P}_{\mathbf{v}}(G)$, measuring the proportion of the elements of a finite group $G$ that are zeros of irreducible characters of $G$, takes (as proved in [12]) only values $\frac{m-1}{m}$, for $1 \leq m \leq 6$, in the…

It is proven that if $G$ is a finite group, then $G^\omega$ has $2^{\mathfrak c}$ dense nonmeasurable subgroups. Also, other examples of compact groups with dense nonmeasurable subgroups are presented.

Tkachenko and Yaschenko [34] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is…

For an irreducible character $\chi$ of a finite group $G$, the codegree of $\chi$ is defined as $|G:\ker(\chi)|/\chi(1)$. In this paper, we determine finite nonsolvable groups with exactly three nonlinear irreducible character codegrees,…

A finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient. The smallest examples of exceptional p-groups have order p^5. For an odd…

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an…

The Alon-Tarsi number $AT(G)$ of a graph $G$ is the smallest $k$ for which there is an orientation $D$ of $G$ with max indegree $k-1$ such that the number of even and odd circulations contained in D are different. In this paper, we show…

Given a finite non-cyclic group $G$, call $\sigma(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $\sigma(G)$ for $G$ a group with a unique minimal normal subgroup $N$…