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Patrick X. Gallagher
June 29, 2012

### Abstract

Abstract. In the character table of a finite group there is a tendency either for the character degree to divide the conjugacy class size or the character value to vanish. There is also a partial divisibility where the determinant of the character is not 1. There are versions of these depending on a subgroup, based on an arithmetic property of spherical functions which generalizes the integrality of the values of the characters and the central characters.

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Mark L. Lewis
June 29, 2012

### Abstract

Abstract. Let be a Camina pair. We prove that G must be a p -group for some prime p . We also prove that . Also, we discuss how one might build examples with , although we are not able to prove the existence of such examples.

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Zoltán Halasi, Attila Maróti, Said Sidki, Marcelo Bezerra
June 29, 2012

### Abstract

Abstract. A finite group G is called expansive if for every normal set S and every conjugacy class C of G the normal set consists of at least as many conjugacy classes of G as S does. This notion is motivated by a finiteness criterion. It is shown that a group is expansive if and only if it is a direct product of expansive simple or abelian groups. The groups and are expansive for every and every respectively. Many small simple groups including all sporadic simple groups are also expansive.

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Sarah B. Hart, Peter J. Rowley
June 29, 2012

### Abstract

Abstract. Let be the set of strongly real elements of a Coxeter group W . Then for , the excess of w is defined by When W is finite we may also define , the reflection excess of w . The main result established here is that if W is finite and X is a W -conjugacy class, then there exists such that w has minimal length in X and .

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Peter Davidson
June 29, 2012

### Abstract

Abstract. Under suitable conditions upper bounds of second order Dehn functions of Pride groups are obtained. From this we show that the second order Dehn function of a right-angled Artin group is at most quadratic.

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Mojtaba Ramezan-Nassab, Dariush Kiani
June 29, 2012

### Abstract

Abstract. Let D be a non-commutative division ring, m and n two natural numbers, M a maximal and N a subnormal subgroup of . In this paper, among other results, we show that: (1) if N is an n -Engel group, then it is central; (2) if and M is locally nilpotent, then M is abelian; (3) if and M is n -Engel, then the Hirsch–Plotkin radical of M is abelian. Also, we define some generalized Engel conditions on groups, and then we prove similar results, as quoted, for these groups.

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Daniel McLaury
June 29, 2012

### Abstract

Abstract. In this paper, we classify the finite-dimensional irreducible linear representations of the Baumslag–Solitar groups for relatively prime p and q . The general strategy is to consider the matrix group given by the image of a representation and study its Zariski closure in .

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Alejandro Adem, Karel Dekimpe, Nansen Petrosyan, Bartosz Putrycz
June 29, 2012

### Abstract

Abstract. We find new bounds on the minimal number of generators of crystallographic groups with p -group holonomy. We also show that similar bounds exist on the minimal number of generators of the abelianizations of arbitrary crystallographic groups. As a consequence, we show that this restricts the rank of elementary abelian p -groups that can act effectively on closed connected flat orbifolds.

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Gustavo A. Fernández-Alcober, Amaia Zugadi-Reizabal
June 29, 2012

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