Some Open Problems on a Class of Finite Groups

AbstractA group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later, in 1921, he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this paper we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.

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On two open problems of the theory of permutable subgroups of finite groups

Publicationes Mathematicae Debrecen ◽

10.5486/pmd.2019.8473 ◽

2019 ◽

Vol 94 (3-4) ◽

pp. 477-491

Author(s):

Bin Hu ◽

Jianhong Huang ◽

Alexander N. Skiba

Keyword(s):

Finite Groups ◽

Open Problems ◽

Permutable Subgroups

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Global and local properties of finite groups with only finitely many central units in their integral group ring

Journal of Group Theory ◽

10.1515/jgth-2020-0165 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Andreas Bächle ◽

Mauricio Caicedo ◽

Eric Jespers ◽

Sugandha Maheshwary

Keyword(s):

Finite Groups ◽

Nilpotent Groups ◽

Galois Groups ◽

Group Rings ◽

Integral Group ◽

Open Problems ◽

Integral Group Rings ◽

Local Properties ◽

Global And Local ◽

The Impact

Abstract The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group, we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same; in particular, the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.

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On ℱ-residuals of finite groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700020311 ◽

2002 ◽

Vol 65 (2) ◽

pp. 271-275

Author(s):

Wenbin Guo ◽

K. P. Shum ◽

A. N. Skiba

Keyword(s):

Finite Groups ◽

Negative Answer ◽

Open Problems ◽

Subdirect Products ◽

Closed Formation

We prove that there exists a soluble, saturated and s-closed formation ℱ of groups such that the class (Gℱ | G is a group, Gℱ is the ℱ-residual of G) is not closed under subdirect products. This result a negative answer to an open problems recently proposed by L.A. Shemetkov in 1998.

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