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Rod Gow, C. Ryan Vinroot
May 21, 2008

### Abstract

Let GL and U denote the finite general linear and unitary groups extended by the transpose inverse automorphism, respectively, where q is a power of the prime p . Let n be odd, and let χ be an irreducible character of either of these groups which is an extension of a real-valued character of GL or U. Let yτ be an element of GL or U such that ( yτ ) 2 is regular unipotent in GL or U, respectively. We show that is prime to p and otherwise. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.

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C. Ryan Vinroot
May 21, 2008

### Abstract

Let G be a finite Coxeter group. Using previous results on Weyl groups, and covering the cases of non-crystallographic groups, we show that G has an involution model if and only if all of its irreducible factors are of type A n , B n , D 2 n +1 , H 3 , or I 2 ( n ).

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Alexander Moretó, Pham Huu Tiep
May 21, 2008

### Abstract

Pálfy proved that given a solvable group G and a set of prime divisors of character degrees of G of cardinality at least 3, there exist two different primes such that pq divides some character degree. The solvability hypothesis cannot be removed from Pálfy's theorem, but we show that the same conclusion holds for arbitrary finite groups if .

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Clara Franchi, Mario Mainardis, Ronald Solomon
May 21, 2008

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Edith Adan-Bante, Helena Verrill
May 21, 2008

### Abstract

Let S n be the symmetric group of degree n where n > 5. Given any non-trivial , we prove that the product of the conjugacy classes and is never a conjugacy class. Furthermore, if n is odd and not a multiple of three, then is the union of at least three distinct conjugacy classes. We also describe the elements in the case when is the union of exactly two distinct conjugacy classes.

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Gustavo A. Fernández-Alcober, Leire Legarreta
May 21, 2008

### Abstract

Let v ( G ) be the number of conjugacy classes of non-normal subgroups of a finite group G . Poland and Rhemtulla [J. Poland and A. Rhemtulla. The number of conjugacy classes of non-normal subgroups in nilpotent groups. Comm. Algebra 24 (1996), 3237–3245.] proved that if G is nilpotent of class c then c − 1 unless G is a Hamiltonian group. The sharpness of this lower bound is a problem about finite p -groups, since , where μ ( G ) denotes the number of normal subgroups of G . In this paper, we show that the bound c − 1 can be substantially improved: if G is a finite p -group and , then p ( k − 1) + 1, unless G is a Hamiltonian group or a generalized quaternion group. In these exceptional cases, G is a 2-group and 2( k − 1).

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B. A. F. Wehrfritz
May 21, 2008

### Abstract

We make a detailed study of the structure of the normalizer of a locally nilpotent subgroup in the multiplicative group of an arbitrary division ring. This generalizes work on nilpotent such subgroups of class 2 dealt with in M. Shirvani's paper [M. Shirvani. On soluble-by-finite subgroups of division algebras. J. Algebra 294 (2005), 255–277.] and the author's paper [B. A. F. Wehrfritz. Normalizers of nilpotent subgroups of division rings. Quart. J. Math. , to appear.]. For the reasons for doing this, see [M. Shirvani. On soluble-by-finite subgroups of division algebras. J. Algebra 294 (2005), 255–277.] and [B. A. F. Wehrfritz. Normalizers of nilpotent subgroups of division rings. Quart. J. Math. , to appear.] and references there.

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R. B. J. T. Allenby
May 21, 2008

### Abstract

It has been conjectured that the Frattini subgroup and the upper (and hence lower) near Frattini subgroup of a generalized free product are contained in the amalgamated subgroup. This paper adds to the results so far obtained by proving, as a corollary to a much more general result, that the conjectured result holds if the amalgamated subgroup is countable.

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Dikran Dikranjan, Dmitri Shakhmatov
May 21, 2008

### Abstract

We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This is the widest class of groups known to date where the answer to the 63-year-old problem of Markov turns out to be positive. We also prove that whether every unconditionally closed subset of G is algebraic or not is completely determined by countable subgroups of G . Essential connections with non-topologizable groups are highlighted.