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A note on semilocal group rings

Angelina Y. M. Chin (2002)

Czechoslovak Mathematical Journal

Let R be an associative ring with identity and let J ( R ) denote the Jacobson radical of R . R is said to be semilocal if R / J ( R ) is Artinian. In this paper we give necessary and sufficient conditions for the group ring R G , where G is an abelian group, to be semilocal.

A note on simple medial quasigroups

K. K. Ščukin (1993)

Commentationes Mathematicae Universitatis Carolinae

A solvable primitive group with finitely generated abelian stabilizers is finite.

A note on solvable vertex stabilizers of s -transitive graphs of prime valency

Song-Tao Guo, Hailong Hou, Yong Xu (2015)

Czechoslovak Mathematical Journal

A graph X , with a group G of automorphisms of X , is said to be ( G , s ) -transitive, for some s ≥ 1 , if G is transitive on s -arcs but not on ( s + 1 ) -arcs. Let X be a connected ( G , s ) -transitive graph of prime valency p ≥ 5 , and G v the vertex stabilizer of a vertex v ∈ V ( X ) . Suppose that G v is solvable. Weiss (1974) proved that | G v | ∣ p ( p - 1 ) 2 . In this paper, we prove that G v ≅ ( ℤ p ⋊ ℤ m ) × ℤ n for some positive integers m and n such that n div m and m ∣ p - 1 .

A note on supersoluble maximal subgroup and theta-pairs.

James C. Beidleman, Howard Smith (1993)

Publicacions Matemàtiques

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.

A note on the continuous extensions of injective morphisms between free groups to relatively fre profinite groups.

Thierry Coulbois, Mark Sapir, Pascal Weil (2003)

Publicacions Matemàtiques

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...

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