A note on solvability of nite groups (original) (raw)

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.

A note on p-nilpotence and solvability of finite groups

Journal of Algebra, 2009

In this note, we first give some examples to show that some hypotheses of some well-known results for a group G to be pnilpotent, solvable and supersolvable are essential and cannot be removed. Second, we give some generalizations of two theorems in [A. Ballester-Bolinches, X. Guo, Some results on p-nilpotence and solubility of finite groups, J. Algebra 228 (2000) 491-496].

Self-normalizing Sylow subgroups

2004

Using the classification of finite simple groups we prove the following statement: Let p > 3 be a prime, Q a group of automorphisms of p-power order of a finite group G, and P a Q-invariant Sylow p-subgroup of G. If C N G (P)/P (Q) is trivial, then G is solvable. An equivalent formulation is that if G has a self-normalizing Sylow p-subgroup with p > 3 a prime, then G is solvable. We also investigate the possibilities when p = 3. Theorem 1.1. Let p be an odd prime and P a Sylow p-subgroup of the finite group G. If p = 3, assume that G has no composition factors of type L 2 (3 f), f = 3 a with a ≥ 1. (1) If P = N G (P), then G is solvable. (2) If N G (P) = P C G (P), then G/O p (G) is solvable. Note that the second result implies the first since it is well known ([7], [1, Lemma 12.1]) that if H is a group of automorphisms of R with gcd(|H|, |R|) = 1 and C R (H) = 1, then R is solvable. We then apply this result to P acting on O p (G). We will say more about this in the next section. If G is a simple group with p ≥ 5, it was an observation of Thompson that this followed quite easily from a result of Glauberman. See [4, Thm. X.8.15]. An easy consequence of the previous theorem (or our proof) is the extension of this result to p = 3. Corollary 1.2. If p is an odd prime and G is a nonabelian finite simple group, then N G (P) = P C G (P). Proof. By the theorem, we need only consider p = 3 and G = L 2 (3 3 a). Then the split torus acts nontrivially on a Sylow 3-subgroup.

On finite products of nilpotent groups

Archiv der Mathematik, 1994

i. Introduetion. A well-known theorem of Kegel [7] and Wielandt [9] states the solubility of every finite group G = AB which is the product of two nilpotent subgroups A and B; see [1], Theorem 2.4.3. In order to determine the structure of these groups it is of interest to know which subgroups of G are conjugate (or at least isomorphic) to a subgroup that inherits the factorization. A subgroup S of the factorized group G = AB is called prefactorized if S = (A c~ S) (B ~ S), it is called factorized if, in addition, S contains the intersection A c~ B. Generally, even characteristic subgroups of G are not prefactorized, as can be seen e.g. from Examples 1 and 2 below.

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].

Influence of the number of Sylow subgroups on solvability of finite groups

Comptes Rendus Mathematique, 2021

Let G be a finite group. We prove that if the number of Sylow 3-subgroups of G is at most 7 and the number of Sylow 5-subgroups of G is at most 1455, then G is solvable. This is a strong form of a recent conjecture of Robati. 2020 Mathematics Subject Classification. 20D10, 20D20, 20F16, 20F19. Funding. The first author is supported by both TU Graz (R-1501000001) and partial funding from the Austrian Science Fund (FWF): P30934–N35, F05503, F05510. He is also at the University of Nigeria, Nsukka (UNN). The research of the second author is supported by Ministerio de Ciencia e Innovación PID−2019−103854GB−100, Generalitat Valenciana AICO/2020/298 and FEDER funds. Manuscript received 4th October 2020, revised and accepted 5th November 2020.

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.

On Sylow Normalizers of Finite Groups

Journal of Algebra and Its Applications, 2014

The paper considers the influence of Sylow normalizers, i.e., normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belong to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.