Journal of Algebra and Its Applications

The following characterizations of p-groups of maximal class are proved: (a) If a p-group of order > p^p+2 contains a subgroup of maximal class and index p, then G possesses at most one normal subgroup of order p^p and exponent p. (b) If the center of any nonabelian epimorphic image of a nonabelian two-generator p-group G is cyclic, then either G ≅ M_pⁿ or G is of maximal class. (c) An 𝒜_n-group G, n > 1, is of maximal class ⇔ all its 𝒜₂-subgroups of minimal order are of maximal class. (iv) If all factors of the lower central series of a nonabelian two-generator p-group are cyclic, then it is either of maximal class or ≅ M_pⁿ. (v) If a nonabelian p-group G is such that any s pairwise non-commuting elements generate a group of maximal class, where s is the fixed member of the set {3, …, p + 1} and p > 2 if s ≠ p + 1, then G is also of maximal class. We also study the noncyclic p-groups containing only one normal subgroup of a given order.

We compute the upper central series for the regular wreath product finite group C ≀ E where C is a cyclic p-group and E is an elementary abelian p-group for some prime p. The notion of a pattern subgroup enables us to describe the upper central series in a natural way.

Let m ≥ 3 be an integer, ζ_m ∈ ℂ a primitive mth root of unity, and K_m the cyclotomic field ℚ(ζ_m). An explicit description of the integral trace form where is the complex conjugate of x is presented. In the case where m is prime, a procedure for finding the minimum of the form subject to x being a nonzero element of a certain ℤ-module in ℤ[ζ_m] is presented.

A ring R is feckly clean provided that for any a ∈ R there exists an element e ∈ R and a full element u ∈ R such that a = e + u, eR(1 - e) ⊆ J(R). We prove that a ring R is feckly clean if and only if for any a ∈ R, there exists an element e ∈ R such that V(a) ⊆ V(e), V(1 - a) ⊆ V(1 - e) and eR(1 - e) ⊆ J(R), if and only if for any distinct maximal ideals M and N, there exists an element e ∈ R such that e ∈ M, 1 - e ∈ N and eR(1 - e) ⊆ J(R), if and only if J-spec(R) is strongly zero-dimensional, if and only if Max(R) is strongly zero-dimensional and every prime ideal containing J(R) is contained in a unique maximal ideal. More explicit characterizations are also discussed for commutative feckly clean rings.

The main aim of this note is to give a criterion for a subgroup-closed formation to be solubly saturated, which we hope may provide a useful proving ground for outstanding questions about this family of formations.

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.

For a group π, we study the quasitriangular structures of π-twisted tensor biproduct Hopf π-coalgebra which is constructed by the first author and Tang. The necessary and sufficient conditions for such π-twisted tensor biproduct Hopf π-coalgebra to admit quasitriangular structures. As applications of our results, some corollaries are given as well.

Let Q be a Buchsteiner loop. We describe the associator calculus in three variables, and show that |Q| ≥ 32 if Q is not conjugacy closed. We also show that |Q| ≥ 64 if there exists x ∈ Q such that x² is not in the nucleus of Q. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that has both nilpotency class 3 and an abelian inner mapping group.

Modules (cogenerated by nonzero their submodules) having nonzero square homomorphism in nonzero submodules are said (prime) weakly compressible (wc). Such modules are semiprime (i.e. they are cogenerated by their essential submodules). For many rings R, including commutative rings, it is proved that wc modules are isomorphic submodules of products of prime modules, and the converse holds precisely when the class of wc modules is enveloping for mod-R or equivalently, every semiprime module is wc. Semi-Artinian rings R have the latter property and the converse is true when R is strongly regular. Duo Noetherian rings over which wc modules form an enveloping class are shown to have a finite number maximal ideals. If R is Morita equivalent to a Dedekind domain, then the class of wc R-modules is enveloping if and only if R is simple Artinian or J(R) ≠ 0.

We study the 2-adic properties for the numbers of involutions in the alternative groups, and give an affirmative answer to a conjecture of Kim and Kim [A combinatorial approach to the power of 2 in the number of involutions, J. Combin. Theory Ser. A117 (2010) 1082–1094]. Some analogous and general results are also presented.

In this paper, we study some algebraic properties of evolution algebras. Moreover, we reduce the study of evolution algebras of permutations to two special types of evolution algebras, idempotents and absolute nilpotent elements of the algebra. We study three-dimensional evolution algebras whose each element of evolution basis has infinite period. In addition, for an evolution algebra with some properties we describe its associative enveloping algebra.

Let H be a subgroup of a finite group G. We use H_sG to denote the S-core of H, that is, the subgroup of G generated by all those subgroups of H which are S-permutable in G. We say that H/H_sG is the S-cofactor of H. In this paper we study G under restrictions on the S-cofactors of some subgroups of G.

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x^-1, y, y^-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C ⊗_L R〚x, y〛[(xy)^-1] and C ⊗_L R[x, x^-1]〚y〛[y^-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

By a proper cover of a finite group G we mean an extension of a nontrivial finite group by G. We study element orders in proper covers of a finite simple group L of Lie type and prove that such a cover always contains an element whose order differs from the element orders of L provided that L is not L₄(q), U₃(q), U₄(q), U₅(2), or ³D₄(2).

For a finite group G, let v(G) denote the number of conjugacy classes of all non-normal subgroups of G, and let π(G) denote the set of primes dividing the order of G. In this note, we shall classify the finite groups G with v(G) ≤ |π(G)|.

We provide a partial answer to the following question: Assume that R is a finite ring of order s such that for every two subsets M and N of cardinalities m and n respectively, there exist x ∈ M and y ∈ N such that xy = yx. What relations among s, m, n guarantee that R is commutative? Also, we give criteria for commutativity of such rings in terms of structure and relations between m and n.

We determine conditions under which a generalized triangular matrix ring has finite reduced rank, in the general torsion-theoretic sense. These are applied to characterize certain orders in Artinian rings, and to show that if each homomorphic image of a ring S has finite reduced rank, then so does the ring of lower triangular matrices over S.

In this paper, we shall introduce thin n-subpolygroups of a given n-polygroup and in this regards, the notion of wreath product of n-polygroups will be studied. Also, double cosets of n-polygroups are investigated and the classical isomorphism theorems of groups are generalized to n-polygroups. The main result of the paper is that a finite n-polygroup is singular if and only if it is a wreath product of n-subpolygroups all of which are thin or generated by an involution or by an idempotent element.