Journal of Algebra and Its Applications

In this paper, we study the class (Ω, ∞) of groups whose every infinite subset contains two distinct elements generating an Ω-group where Ω is either the class of Černikov groups, or the class of Černikov-by-nilpotent groups and we deduce some characterizations of finite-by-nilpotent groups.

A simple direct method of constructing a faithful, irreducible representation of S_n of degree n - 1 is given.

The problem of determining a complete set of invariants for characterizing the Clifford class of a central simple G-algebra over a field K is studied under the assumption that the algebra is simple ring. This is successful in the case of interior central simple G-algebras over K, and also when the inertia subgroup of the G-algebra is a direct factor of G. As an application of the latter, it is shown that if G is a finite abelian group, L is a Galois extension of K with Galois group G, and E is a G-algebra isomorphic to End_KV for some KG-module V, then the skew group ring of G over L ⊗_K E is a split matrix algebra over K.

Let * denote the canonical involution of the group algebra KG induced by the map x ↦ x^-1 for x ∈ G. In case K is a real extension of ℚ, we consider Cayley unitary elements built out of skew elements k = α(x - x^-1) in KG such that 1 + k is invertible in KG, for α ∈ K and x ∈ G. The constructions involve an interesting sequence in the coefficients of (1 + k)^-1 which is the Fibonacci sequence when α = 1.

For a finite dimensional monomial algebra Λ over a field K we show that the Hochschild cohomology ring of Λ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated K-algebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field in [13].

Let R be an integral domain in which 2 is not an invertible element, with quotient field K of characteristic not 2. A construction method for octonion algebras over Ris presented for which the resulting algebra does not necessarily contain a composition subalgebra.

If (R, M) and (S, N) are quasi-local domains and f : R → S is a ring homomorphism, then f is said to be a strong local homomorphism if f(M) = N. Let PVD be the category whose objects are the pseudo-valuation domains that are not fields and whose morphisms are the strong local homomorphisms; let VD be the full subcategory of PVD whose objects are all the valuation domains that are not fields. Then VD is shown to be a reflective subcategory of PVD. This result is extended by obtaining a reflective conclusion for a category whose class of objects properly contains all pseudo-valuation domains.

In this paper, we reexamine the foundation of Isaacs' π-theory. One of the key concepts in Isaacs' π-theory is the construction of the characters B_π(G) for a π-separable group G. The key to determining which characters lie in B_π(G) was the construction of a nucleus for each irreducible character χ. In this paper, we present a different way of finding a nucleus for χ which is based on a chain of normal subgroup . Using this nucleus, we obtain the set of characters . We investigate the properties that has in common with B_π(G).

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Z_m, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved.

In this paper, we extend Hoit's results by replacing the Abelian group (Z_m, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.