Shi’s Conjecture

In this report we summarize this work, all finite simple groups G can determined uniformly using their orders |G| and the set π e (G) of their element orders.

2010

Let G be a finite group and let OC(G) be the set of order components of G. The number of non-isomorphic classes of finite groups H satisfying OC(G) = OC(H) is denoted by h(G). If h(G) = k then G is called a k-recognizable group by the set of its order components and if k = 1, G is called a recognizable group. The main consequence of recognizability of a group G by its order components is the validity of Thompson’s conjecture for G. In this paper we consider recognizability of simple groups with exactly two components and we survey currently known results in this regard which shows that the recognition problem for these groups is completely solved.

2018

In this note we provide some counterexamples for the conjectures of finite simple groups, one of the conjectures said "all finite simple groups $G$ can be determined using their orders $|G|$ and the number of elements of order $p$, where $p$ the largest prime divisor of $|G|$".

Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021

In this note we provide some counterexamples for the conjecture of Moretó on finite simple groups, which says that any finite simple group G can be determined in terms of its order |G| and the number of elements of order p, where p the largest prime divisor of |G|. A new characterization of all sporadic simple groups and alternating groups is given. Some related conjectures are also discussed.

arXiv: Group Theory, 2020

In this note we provide some counterexamples for the conjecture of Moret\'{o} on finite simple groups, which says that any finite simple group $G$ can determined in terms of its order $|G|$ and the number of elements of order $p$, where $p$ the largest prime divisor of $|G|$. Moreover, we show that this conjecture holds for all sporadic simple groups and alternating groups $A_n$, where $n\neq 8, 10$. Some related conjectures are also discussed.

Communications in Algebra, 2000

We s h o w that all but A 6 non-abelian nite simple groups with no elements of order 6 are characterized by their element orders.

2021

The spectrum of a finite group is the set of element orders of this group. The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum, in particular, to list all finite simple groups for which the recognition problem is solved.

Journal of Mathematics Research, 2021

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1 and p_2 are two different primes. We also show that for a given different prime numbers p and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

2013

In this report we summarize this work, all finite simple groups G can determined uniformly using their orders |G| and the set π e (G) of their element orders.

Proceedings of the American Mathematical Society, 1965