Another criterion for solvability of finite groups (original) (raw)

A sufficient conditon for solvability of finite groups

arXiv (Cornell University), 2017

The following theorem is proved: Let G be a finite group and π e (G) be the set of element orders in G. If π e (G) ∩ {2} = ∅; or π e (G) ∩ {3, 4} = ∅; or π e (G) ∩ {3, 5} = ∅, then G is solvable. Moreover, using the intersection with π e (G) being empty set to judge G is solvable or not, only the above three cases. 1 Introduction Let G be a finite group. We have two basic sets: |G| and π e (G). There are many famous works about |G| in the history of group theory. The set π e (G)

A result on the sum of element orders of a finite group

Archiv der Mathematik, 2019

Let G be a finite group and ψ(G) = g∈G o(g). There are some results about the relation between ψ(G) and the structure of G. For instance, it is proved that if G is a group of order n and ψ(G) > 211 1617 ψ(C n), then G is solvable. Herzog et al. in [Herzog et al., Two new criteria for solvability of finite groups, J. Algebra, 2018] put forward the following conjecture: Conjecture. If G is a non-solvable group of order n, then ψ(G) ≤ 211 1617 ψ(C n) with equality if and only if G = A 5. In particular, this inequality holds for all non-abelian simple groups. In this paper, we prove a modified version of Herzog's Conjecture.

A new solvability criterion for finite groups

2010

In 1968, John Thompson proved that a finite group GGG is solvable if and only if every 222-generator subgroup of GGG is solvable. In this paper, we prove that solvability of a finite group GGG is guaranteed by a seemingly weaker condition: GGG is solvable if for all conjugacy classes CCC and DDD of GGG, \emph{there exist} xinCx\in CxinC and yinDy\in DyinD for which genx,y\gen{x,y}genx,y is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if GGG is a finite nonabelian simple group, then there exist two integers aaa and bbb which represent orders of elements in GGG and for all elements x,yinGx,y\in Gx,yinG with ∣x∣=a|x|=a∣x∣=a and ∣y∣=b|y|=b∣y∣=b, the subgroup genx,y\gen{x,y}genx,y is nonsolvable.

A note on the solvability of groups

Journal of Algebra, 2006

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K M. We call the group M ∩ K/L a c-section of M. And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M. For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L 2 (p) or Z q , where p and q are primes, and p ≡ ±1(mod 8). This result answer a question posed by ref. [12].

The minimum sum of element orders of finite groups

International Journal of Group Theory, 2019

Let G be a finite group and ψ(G) = ∑ g∈G o(g), where o(g) denotes the order of g ∈ G. We show that the Conjecture 4.6.5 posed in [Group Theory and Computation, (2018) 59-90], is incorrect. In fact, we find a pair of finite groups G and S of the same order such that ψ(G) < ψ(S), with G solvable and S simple.

A Note on the Solvablity of Groups

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K M. We call the group M ∩ K/L a c-section of M. And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M. For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L 2 (p) or Z q , where p and q are primes, and p ≡ ±1(mod 8). This result answer a question posed by ref. [12].

An exact upper bound for sums of element orders in non-cyclic finite groups

Journal of Pure and Applied Algebra, 2017

Denote the sum of element orders in a finite group G by ψ(G) and let C n denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that ψ(G) ≤ 7 11 ψ(C n) and ψ(G) < 1 q−1 ψ(C n). The first result is best possible, since for each n = 4k, k odd, there exists a group G of order n satisfying ψ(G) = 7 11 ψ(C n) and the second result implies that if G is of odd order, then ψ(G) < 1 2 ψ(C n). Our results improve the inequality ψ(G) < ψ(C n) obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some ψ(G)-based sufficient conditions for the solvability of G.

A solvability criterion for finite groups related to the number of Sylow subgroups

Communications in Algebra, 2020

Let G be a finite group and let pðGÞ be the set of primes dividing the order of G. For each p 2 pðGÞ, the Sylow theorems state that the number of Sylow p-subgroups of G is equal to kp þ 1 for some non-negative integer k. In this article, we characterize non-solvable groups G containing at most p 2 þ 1 Sylow p-subgroups for each p 2 pðGÞ: In particular, we show that each finite group G containing at most ðp À 1Þp þ 1 Sylow p-subgroups for each p 2 pðGÞ is solvable.