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George Glauberman
September 8, 2005

### Abstract

Suppose p is a prime, S is a finite p -group, and A is an abelian subgroup of S . Does S possess a normal abelian subgroup of the same order as A ? J. Alperin showed that the answer is negative in general. However, by using methods of algebraic group theory, he and the author showed that the answer is affirmative if | A | = p n , where n is ‘small’ relative to p . In this paper, we prove the dual result that the answer is affirmative if the index | S : A | is ‘small’ relative to p . We also prove a correspondence between the abelian subgroups of a p -group of nilpotence class at most p and the abelian subgroups of a related p -group of nilpotence class at most two.

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Mark L. Lewis, Alexander Moretó, Thomas R. Wolf
September 8, 2005

### Abstract

In this paper, we study groups for which if 1 < a < b are character degrees, then a does not divide b . We say that these groups have the condition no divisibility among degrees (NDAD). We conjecture that the number of character degrees of a group that satisfies NDAD is bounded and we prove this for solvable groups. More precisely, we prove that solvable groups with NDAD have at most four character degrees and have derived length at most 3. We give a group-theoretic characterization of the solvable groups satisfying NDAD with four character degrees. Since the structure of groups with at most three character degrees is known, these results describe the structure of solvable groups with NDAD.

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Peter Schmid
September 8, 2005

### Abstract

Let p be a rational prime. The k ( GV ) theorem states that, given any finite p' -group G and any finite faithful -module V , the number of conjugacy classes of the semidirect product GV is bounded from above by the order of V (that is, k ( GV ) ≤ | V |). When do we have equality k ( GV ) = | V |?

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A. Ballester-Bolinches, L. M. Ezquerro
September 8, 2005

### Abstract

We say that a formation ℱ of finite groups has the Kegel property if ℱ contains every group of the form G = AB = BC = CA with A , B , C in ℱ. Vasil’ev asked the following question in the Kourovka Notebook: if ℱ is a soluble Fitting formation of finite groups with the Kegel property must ℱ be a saturated formation? We obtain an affirmative answer in the soluble universe in the case when ℱ has the following additional property: for every prime p ∈ char ℱ and every primitive ℱ-group G whose socle is a p -group, lies in ℱ for all primes q ≠ p such that q divides | G | Soc( G )|.

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Gregory Cherlin
September 8, 2005

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Olivier Frécon, Eric Jaligot
September 8, 2005

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Haryono Tandra, William Moran
September 8, 2005

### Abstract

Extensions of characters of a discrete, torsion-free, nilpotent group to characters of some of its supergroups in its Mal’tsev completion are examined. Some sufficient conditions, as well as some necessary conditions, for extensibility are provided. In particular, the situation for nilpotent groups of class 2 is discussed.

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Dessislava H. Kochloukova
September 8, 2005

### Abstract

We establish a sufficient condition for some modules M over the group algebra ℤ[ G ] to be of homological type FP 2 , where G is a finitely generated split extension of abelian groups. This generalizes a result of Bieri and Strebel [R. Bieri and R. Strebel. Valuations and finitely presented metabelian groups. Proc. London Math. Soc. (3) 41 (1980), 439–464] when M is the trivial module ℤ and it establishes a special case of [D. H. Kochloukova. A new characterisation of m -tame groups over finitely generated abelian groups. J. London Math. Soc. (2) 60 (1999), 802–816, Conjecture K. S. Brown. Cohomology of groups (Springer-Verlag, 1982)].