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

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].

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 supersolvable groups whose maximal subgroups of the Sylow subgroups are subnormal

Revista de la Unión Matemática Argentina, 2019

A finite group G is called an MSN *-group if it is supersolvable, and all maximal subgroups of the Sylow subgroups of G are subnormal in G. A group G is called a minimal non-MSN *-group if every proper subgroup of G is an MSN *-group but G itself is not. In this paper, we obtain a complete classification of minimal non-MSN *-groups.

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)

Schreier Conditions on Chief Factors and Residuals of Solvable-Like Group Formations

Bulletin of the Australian Mathematical Society, 2008

Let α be a formation of finite groups which is closed under subgroups and group extensions and which contains the formation of solvable groups. Let G be any finite group. We state and prove equivalences between conditions on chief factors of G and structural characterizations of the α-residual and theα-radical of G. We also discuss the connection of our results to the generalized Fitting subgroup of G.

New Trends in Characterization of Solvable Groups

webdoc.sub.gwdg.de

Abstract. We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for ...