Volume 15, Issue 3

Abstract. We determine the structure of 2-blocks with minimal nonabelian defect groups, by making use of the classification of finite simple groups.

Abstract. Let be a defect group of a 2-block of a finite group. We conjecture that if is rational and of nilpotence class at most 2, then the values of every character in lie in a cyclotomic field , for some odd integer . We prove the conjecture when has maximal defect.

Abstract. For a finite group , let indicate the minimum number of generators of . A (finite) group is monotone if, for every pair of subgroups of , implies . Monotone -groups have been introduced and investigated by A. Mann, who determined their structure if . In this paper, a complete classification of monotone 2-groups is given.

Abstract. Let be a subgroup of a group . In this paper, we say that is -supplemented in if there is a subgroup of such that and , where denotes the subgroup of generated by all those subgroups of which are -quasinormally embedded in . We strengthen a nice result of Skiba which gives a positive answer to an open question of Shemetkov.

Abstract. Let be a finite group of order , where is a prime and is not divisible by , and let be a Sylow -subgroup of . If is an outer commutator word, we prove that is generated by the intersection of with the set of th powers of all values of in .

Abstract. For a topological group the intersection of all kernels of ordinary representations is studied. We show that is contained in the center of  if is a connected pro-Lie group. The class is determined explicitly if is the class of connected Lie groups or the class of almost-connected Lie groups: in both cases, it consists of all compactly-generated abelian Lie groups. Every compact abelian group and every connected abelian pro-Lie group occurs as for some connected pro-Lie group . However, the dimension of is bounded by the cardinality of the continuum if is locally compact and connected. Examples are given to show that becomes complicated if contains groups with infinitely many connected components.

Abstract. We prove a structure theorem for the fundamental group of the quotient of a product of curves by the action of a finite group and, hence, for that of any resolution of the singularities of .