Volume 15, Issue 4

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.

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.

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.

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 .

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.

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.

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 .

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.