Let be an associative ring with identity and let denote the Jacobson radical of . is said to be semilocal if is Artinian. In this paper we give necessary and sufficient conditions for the group ring , where is an abelian group, to be semilocal.
A solvable primitive group with finitely generated abelian stabilizers is finite.
A graph , with a group of automorphisms of , is said to be -transitive, for some , if is transitive on -arcs but not on -arcs. Let be a connected -transitive graph of prime valency , and the vertex stabilizer of a vertex . Suppose that is solvable. Weiss (1974) proved that . In this paper, we prove that for some positive integers and such that and .
A θ-pair for a maximal subgroup M of a group G is a pair (A, B) of subgroups such that B is a maximal G-invariant subgroup of A with B but not A contained in M. θ-pairs are considered here in some groups having supersoluble maximal subgroups.
Let V be a pseudovariety of finite groups such that free groups are residually V, and let φ: F(A) → F(B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension φ' of φ between the pro-V completions of F(A) and F(B) is also injective. In particular, if V is extension-closed, this is the case if and only if φ(F(A)) and its pro-V closure in F(B) have the same rank. We examine a number of situations where the injectivity of φ' can...