相关论文: Proportions of vanishing elements in finite groups

For a finite non-abelian group $G$ let $\rat(G)$ denote the largest ratio of degrees of two nonlinear irreducible characters of $G$. We prove that the number of non-abelian composition factors of $G$ is bounded above by…

For an element $g$ in a group $X$, we say that $g$ has 2-part order $2^{a}$ if $2^{a}$ is the largest power of 2 dividing the order of $g$. We prove lower bounds on the proportion of elements in finite classical groups in odd characteristic…

An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those…

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…

In this paper we investigate the minimum number of maximal subgroups H_i for i=1 ...k of the symmetric group S_n (or the alternating group A_n) such that each element in the group S_n (respectively A_n) lies in some conjugate of one of the…

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many…

Let $G$ be a finite group and $\pi$ be a set of primes. We study finite groups with a large number of conjugacy classes of $\pi$-elements. In particular, we obtain precise lower bounds for this number in terms of the $\pi$-part of the order…

We obtain an asymptotic upper bound for the product of the $p$-parts of the orders of certain composition factors of a finite group acting completely reducibly and faithfully on a finite vector space of order divisible by a prime $p$. An…

We show that there is a class of finite groups, the so-called perfect groups, which cannot exhibit anomalies. This implies that all non-Abelian finite simple groups are anomaly-free. On the other hand, non-perfect groups generically suffer…

We find a lower bound to the size of finite groups detecting a given word in the free group, more precisely we construct a word w_n of length n in non-abelian free groups with the property that w_n is the identity on all finite quotients of…

Let $o(G)$ be the average order of a finite group $G$. We show that if $o(G)<c$, where $c\in \lbrace \frac{13}{6}, \frac{11}{4}\rbrace$, then $G$ is an elementary abelian 2-group or a solvable group, respectively. Also, we prove that the…

A set of quasi-uniform random variables $X_1,...,X_n$ may be generated from a finite group $G$ and $n$ of its subgroups, with the corresponding entropic vector depending on the subgroup structure of $G$. It is known that the set of entropic…

We study the limiting behavior of extremal cohomology groups of $k$-points configuration spaces of complex projective spaces of complex dimension $m\geq 4.$ In the previous work, we prove that the extremal cohomology groups of degrees…

Let $\rho(n)$ denote the maximal number of different primes that may occur in the order of a finite solvable group $G$, all elements of which have orders divisible by at most $n$ distinct primes. We show that $\rho(n)\leq 5n$ for all $n\geq…

We determine the groups of minimal order in which all groups of order n can embedded for 1 < n < 16. We further determine the order of a minimal group in which all groups or order n or less can be embedded, also for 1 < n < 16.

We find a lower bound on the proportion of derangements in a finite transitive group that depends on the minimal nontrivial subdegree. As a consequence, we prove that, if $\Gamma$ is a $G$-vertex-transitive digraph of valency $d\ge 1$, then…

Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $\ell$-algebras, preferably in $f$-algebras, and in product algebras. The additional structure of an associative multiplication leads…

It is well-known that random groups with at least as many relators as generators have vanishing first betti number with high probability. In this paper we extend this question and study maps from few-relators random groups to infinite…

In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which…

We study the probability that certain laws are satisfied on infinite groups, focusing on elements sampled by random walks. For several group laws, including the metabelian one, we construct examples of infinite groups for which the law…