GBRDs over supersolvable groups and solvable groups of order prime to 3 (original) (raw)

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].

A Note on the Solvablity of Groups

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 ...

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 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)

Varieties of finite supersolvable groups with the M. Hall property

Mathematische Annalen, 2006

The varieties in the title are shown to be precisely the product varieties G p * Ab(d) for some prime p and some positive integer d dividing p − 1. Here G p denotes the variety of all finite p-groups and Ab(d) the variety of all finite Abelian groups of exponent dividing d. It turns out that these are exactly those varieties H of supersolvable groups for which all finitely generated free pro-H groups are freely indexed in the sense of Lubotzky and van den Dries. Several alternative characterizations of these varieties are presented. Some applications to formal language theory and finite monoid theory are also given. Among these is the determination of all supersolvable solutions H to the equations PH = J * H and J * H = J m H which is, to the present date, the most complete solution to a problem raised by Pin. Another consequence of our results is that for each such variety H the monoid variety PH = J * H = J m H has decidable membership.

GBRDs over supersolvable groups and solvable groups of order prime to 3 (original) (raw)

Related papers