Groups with two Sylow numbers are the product of two nilpotent Hall subgroups

We have recently found a characterization of the existence of nilpotent Hall subgroups in a finite group. Using this result, we prove the statement in the title. This is a strong form of a conjecture of J. Zhang.     

Generalized evaluation subgroups of product spaces relative to a factor

For any -complexes and , we show that . We use this fact to compute generalized evaluation subgroups of generalized tori relative to a sphere.

     

Groups with maximal subgroups of Sylow subgroups normal

This paper characterizes those finite groups with the property that maximal subgroups of Sylow subgroups are normal. They are all certain extensions of nilpotent groups by cyclic groups.     

Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

Groups isomorphic to all their non-trivial normal subgroups

A finite collectionP of finite sets tiles the integers iff the integers can be expressed as a disjoint union of translates of members ofP. We associate with such a tiling a doubly infinite sequence with entries fromP. The set of all such sequences is a sofic system, called a tiling system. We show that, up to powers of the shift, every shift of finite type can be realized as a tiling system. Some of this work was done at the Mathematical Sciences Research Institute (MSRI), where research is supported in part by NSF grant DMS-9701755. The first two authors thank K. Schmidt for useful conversations and ideas.     

Pronormal subgroups of a direct product of groups

We give criteria to characterize abnormal, pronormal and locally pronormal subgroups of a direct product of two finite groups $A\times B$, under hypotheses of solvability for at least one of the factors, either A or B.     

On the number of non conjugate subgroups

We shall prove that every group of cardinality ₁ has at least ₁ non conjugate subgroups, and we shall generalize this theorem to many more uncountable cardinalities. For example underGCH for every uncountable cardinal and every groupG of cardinality ,G has at least non conjugate subgroups.Presented by W. Taylor.I would like to thank Rami Grossberg for writing and rewriting this paper, and Wilfrid Hodges for removing many errors and suggesting improvements in presentation; many facts are proved only due to his explicit request.This research was supported by grant (No. 1110) from the United States-Israel Binational Science Foundation.     

Groups with many normal-by-finite subgroups

A subgroup of a group is said to be normal-by-finite if the core of in has finite index in . In this article groups satisfying the minimal condition on subgroups which are not normal-by-finite and groups with finitely many conjugacy classes of subgroups which are not normal-by-finite are characterized.

     

Groups with dense normal subgroups

A group is said to have dense normal subgroups, if each non-empty open interval in its lattice of subgroups contains a normal subgroup. The structure of this and related classes of groups is investigated. Typical results are: an infinite group with dense ascendant subgroups is locally nilpotent: a nontorsion group with dense normal subgroups is abelian, etc.     

Groups with pronormal primary subgroups

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 323–327, March, 1989.     

Groups with complemented nonmodular subgroups

We give the structure of finite groups belonging to some families defined by conditions of complementation.