Maximal Order of an NG-group

This study was aimed to consider the NG-group that consisting of transformations on a nonempty set A has no bijection as its element. In addition, it tried to find the maximal order of these groups. It found the order of NG-group not greater than n. Our results proved by showing that any kind of NG-group in the theorem be isomorphic to a permutation group on a quotient set of A with respect to an equivalence relation on A. Keywords: NG-group; Permutation group; Equivalence relation; -subgroup

A new characterization of PSL(3,q)

Let G be a group. In this paper, we prove that G is isomorphic to PSL(3,q) if and only if |G|=|PSL(3,q)| and m(G)=m(PSL(3,q)), where q is a prime power and m(G) is the maximal order of elements in G.

Finite groups with 4p 2 q elements of maximal order

It is an interesting and difficult topic to determine the structure of a finite group by the number of elements of maximal order. This topic is related to Thompson’s conjecture, that is, if two finite groups have the same order type and one of them is solvable, then the other is solvable. In this article, we continue this work and prove that if G G is a finite group which has 4 p 2 q 4{p}^{2}q elements of maximal order, where p p , q q are primes and 7 ≤ p ≤ q 7\le p\le q , then either G G is solvable or G G has a section who is isomorphic to one of L 2 ( 7 ) {L}_{2}\left(7) , L 2 ( 8 ) {L}_{2}\left(8) or U 3 ( 3 ) {U}_{3}\left(3) .

Non-vanishing of class group L-functions for number fields with a small regulator

Let $E/\mathbb {Q}$ be a number field of degree $n$. We show that if $\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is $\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.

Maxmal Order of NG-Transformation Group

In this paper, we consider the problem that the maximal order consider the groups that consisting of transformations we called NG-Transformation on a nonempty set A has no bijection as its element. We find the order of these groups not greater that (n-1)!. In addition, we will prove our result by showing that any kind of NG-group in the theorem be isomorphic to a permutation group on a quotient set of A with respect to an equivalence relation on A.

Maxmal Order of NG-Transformation

In this paper, we consider the problem that the maximal order consider the groups that consisting of transformations we called NG-Transformation on a nonempty set A has no bijection as its element. We find the order of these groups not greater that (n-1)!. In addition, we will prove our result by showing that any kind of NG-group in the theorem be isomorphic to a permutation group on a quotient set of A with respect to an equivalence relation on A.