Pos Groups Revisited

Arxiv preprint arXiv:1007.0568, 2010

If G is a finite group and x ∈ G then the set of all elements of G having the same order as x is called an order subset of G determined by x (see [2]). We say that G is a group with perfect order subsets or briefly, G is a P OS-group if the number of elements in each order subset of G is a divisor of |G|. In this paper we prove that for any n ≥ 4, the symmetric group S n is not P OS-group. Together with the result in [1], this gives the complete positive answer to Conjecture 5.2 in [3].

arXiv: Group Theory, 2020

In this paper we study the finite groups in which every element has prime power order, briefly them EPPO-groups. The classification of EPPO-groups is given including the cases of solvable, non-solvable and simple EPPO-groups. This paper is published in Journal of Yunnan Education College, no.1(1986), p.2-10 (in Chinese). Translate it to English is helpful for readers for citing some conclusions of this paper. For example, the result of solvable EPPO-groups(see Theorem 2.4 in the text) is detailed more than G. Higman's conclusion (see reference 1 in this paper).

arXiv (Cornell University), 2022

In this paper we shall deal with periodic groups, in which each element has a prime power order. A group G will be called a BCP -group if each element of G has a prime power order and for each p ∈ π(G) there exists a positive integer u p such that each p-element of G is of order p i ≤ p u p . A group G will be called a BSP -group if each element of G has a prime power order and for each p ∈ π(G) there exists a positive integer v p such that each finite p-subgroup of G is of order p j ≤ p v p . Here π(G) denotes the set of all primes dividing the order of some element of G. Our main results are the following four theorems. Theorem 1: Let G be a finitely generated BCP -group. Then G has only a finite number of normal subgroups of finite index. Theorem 4: Let G be a locally graded BCP -group. Then G is a locally finite group. Theorem 7: Let G be a locally graded BSP -group. Then G is a finite group. Theorem 9: Let G be a BSP -group satisfying 2 ∈ π(G). Then G is a locally finite group.

Israel journal of mathematics, 1993

Let n be a positive integer or infinity (denote ∞). We denote by W * (n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X 0 ⊆ X, with 2 ≤ |X 0 | ≤ n + 1 and a function f : {0, 1, 2,. .. , k} −→ X0, with f (0) = f (1) and non-zero integers t0, t1,. .. , t k such that [x t 0 0 , x t 1 1 ,. .. , x t k k ] = 1, where xi := f (i), i = 0,. .. , k, and xj ∈ H whenever x t j j ∈ H, for some subgroup H = D x t j j E of G. If the integer k is fixed for every subset X we obtain the class W * k (n). Here we prove that (1) Let G ∈ W * (n), n a positive integer, be a finite group, p > n a prime divisor of the order of G, P a Sylow p-subgroup of G. Then there exists a normal subgroup K of G such that G = P × K. (2) A finitely generated soluble group has the property W * (∞) if and only if it is finite-by-nilpotent. (3) Let G ∈ W * k (∞) be a finitely generated soluble group, then G is finite-by-(nilpotent of k-bounded class).

The American Mathematical Monthly, 2019

In this note we show that a group with a finite number of elements of maximal order must be finite. In other words, there are no infinite groups with finitely many elements of maximal order. In particular, we give an explicit bound for the order of a group in terms of the number of elements of maximal order.

Bulletin of the Australian Mathematical Society, 2014

The classes of finite groups with minimal sets of generators of fixed cardinalities, named B-groups, and groups with the basis property, in which every subgroup is a B-group, contain only p-groups and some {p, q}-groups. Moreover, abelian B-groups are exactly p-groups. If only generators of prime power orders are considered, then an analogue of property B is denoted by B pp and an analogue of the basis property is called the pp-basis property. These classes are larger and contain all nilpotent groups and some cyclic q-extensions of p-groups. In this paper we characterise all finite groups with the pp-basis property as products of p-groups and precisely described {p, q}-groups.

Let G be a finite group and let πe(G) be the set of element orders G. Let k ∈ πe(G) and let m k be the number of elements of order k in G. Set nse(G):={m k |k ∈ πe(G)}. In this paper, we prove that if G is a group such that nse(G)=nse(Sn) where n ∈ {3, 4, 5, 6, 7}, then G ∼ = Sn.

A finite group G is said to have Perfect Order Subsets if for every d, the number of elements of G of order d (if there are any) divides |G|. Answering a question of Finch and Jones from 2002, we prove that if G is Abelian, then such a group has order divisible by 3 except in the case G = Z/2Z. We also place additional restrictions on the order of such groups.

Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali, 2018

In this article we introduce the notion of e-group as a new generalization of a group. The condition for a group to be an e-group is given. The characterization of some properties is established and some results follow. * 2 a b c d a a a a a b a b c d c c c a d d b d b c Then (G; * 1 , A) satisfies (G1) and (eG2) but does not satisfy (eG3). In addition, (G; * 2 ; A) satisfies (eG2) and (eG3) but does not satisfy (G1), since

International Journal of Mathematics and Mathematical Sciences, 2003

We suppose that p = 2 α 3 β + 1, where α ≥ 1, β ≥ 0, and p ≥ 7 is a prime number. Then we prove that the simple groups A n ,

Modnet preprint

An ordered structure M is called o-λ-stable if for any subset A with |A| ≤ λ and for any cut in M there are at most λ 1-types over A which are consistent with the cut. It is proved in the paper that an ordered o-stable group is abelian. Also there were investigated definable subsets and unary functions of o-stable groups.

2003

In the first part of the paper we give a characterization of groups generated by elements of fixed prime order p. In the second part we study the group G (p) n of n×n matrices with the p-th power of the determinant equal to 1 over a field F containing a primitive p-th root of 1. It is known that the group G (2) n of n × n matrices of determinant +−1 over a field F and the group SLn(F ) are generated by their involutions and that each element in these groups is a product of four involutions. We consider some subgroups G of G (p) n and study the following problems: Is G generated by its elements of order p ? If so, is every element of G a product of k elements of order p for some fixed integer k ? We show that G (p) n and SLn(F ) are generated by their elements of order p and that the bound k exists and is equal to 4. We show that every universal p-Coxeter group has faithful two-dimensional representations over many fields F (including R and C). For a universal p-Coxeter group of rank...

Journal of Algebra, 2016

Proof. If |S| = 5, then by Theorem D, |S| ≥ 13. So it suffices to assume that |S| = 13 and to reach a contradiction. Write S = {x 1 , x 2 , x 3 , x 4 , x 5 }, x 1 < x 2 < x 3 < x 4 < x 5 , T = {x 1 , x 2 , x 3 , x 4 } and suppose that |S 2 | = 13. Arguing as in the previous proposition, we may conclude that |T 2 | = 10, |S 2 | = |T 2 | + 3 and T = G. Hence T satisfies the hypotheses of part (iii) of Theorem 2. Suppose first that T = {t, tc, tc t , tc t 2 }, with c > 1 and c t 2 = cc t = c t c. Then x 1 = t, x 2 = tc, x 3 = tc t , x 4 = tc t 2 and as shown in the proof of Lemma 1, c, c t ∈ G ′ and t / ∈ G ′. Moreover, T 2 = {t 2 , t 2 c, t 2 c t , t 2 cc t , t 2 c t c t , t 2 cc t c t , t 2 c 2 c t , t 2 c 2 c t c t , t 2 c(c t) 3 , t 2 c 2 (c t) 3 }, and in particular (3) if t 2 c α (c t) β ∈ T 2 , then α ∈ {0, 1, 2}, β ∈ {0,

Journal of Information and Optimization Sciences, 2020

Given a finite group G, we denote by y ¢(G) the integer (), g G o g ∈ ∏ where o(g) denotes the order of g OE G. It was proved by B. Khosravi and M. Baniasad Azad [12] that some finite simple group can be uniquely determined by its product of element orders. In this paper, we characterize Alt(5) × Z 2 by its order and the product of element orders.

Journal of Algebra, 1976