Algebra Colloquium

Finite p-groups whose subgroups of given order are isomorphic and minimal non-abelian are classified. In addition, two results on a chain condition of 𝒜_t-groups are improved.

For an infinite set X, denote by Ω(X) the semigroup of all surjective mappings from X to X. We determine Green’s relations in Ω(X), show that the kernel (unique minimum ideal) of Ω(X) exists and determine its elements and cardinality. For a countably infinite set X, we describe the elements of Ω(X) for which the 𝒟-class and 𝒥-class coincide. We compare the results for Ω(X) with the corresponding results for other transformation semigroups on X.

Let F, N, A and N₂ denote the properties of being finite, nilpotent, abelian and nilpotent of classes at most 2, respectively. Firstly we consider the class of finitely generated FN-groups. We show that the property FC is closed under finite extensions, and extend this result to finitely generated NF-groups. Secondly we prove that a finitely generated NF-group G is in the class ((FC)F, ∞) if and only if G is an FA-group. Finally we prove that a finitely generated NF-group in the class ((FC)F, ∞)* is an FN₂-group. Moreover, G/Z₂(G) is finite.

The k-characters are class functions on subsets of G^k called the k-classes of G. For a finite group G, the k-class sums form a basis for a subring of ℂG^k, which we call the k-S-ring of G, and we think of these rings as generalized centralizer rings. We show that for a finite group G the 3-S-ring determines G. More specifically, the group characters and set products of certain 3-classes of G, which we call “uniform 3-classes”, determine G.

Bokut, Chen and Huang proved that every countably generated L-algebra over a countable field can be embedded into a simple two-generated L-algebra. In this paper, we prove that every countably generated L-algebra can be embedded into a simple two-generated L-algebra. We also prove that every anti-commutative algebra can be embedded into a simple anti-commutative algebra, and that every countably generated anti-commutative algebra can be embedded into a simple two-generated anti-commutative algebra. Finally, we prove that every anti-commutative algebra can be embedded into its universal enveloping non-associative algebra.

We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.

Let K be a 2-torsion free ring with identity and R_n(K, J) be the ring of all n × n matrices over K such that the entries on and above the main diagonal are elements of an ideal J of K. We describe all Jordan derivations of the matrix ring R_n(K, J) in this paper. The main result states that every Jordan derivation Δ of R_n(K, J) is of the form Δ = D + Ω, where D is a derivation of R_n(K, J) and Ω is an extremal Jordan derivation of R_n(K, J).

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let Φ(x, y) = [x, y]^t – [x, y]^m [α([x, y]),[x, y]]ⁿ [x, y]^s. Thus, (a) if Φ(u, v) = 0 for any u, v ∈ L, then L ⊆ Z(R); (b) if Φ(u, v) ∈ Z(R) for any u, v ∈ L, then either L ⊆ Z(R) or R satisfies s₄, the standard identity of degree 4. We also extend the results to semiprime rings.

Let G be a finite abelian p-group, Γ the maximal ℤ-order of ℤ[G]. We prove that the 2-primary torsion subgroups of K₂(ℤ[G]) and K₂(Γ) are isomorphic when p ≡ 3, 5, 7 (mod 8), and $K_2(\text{ℤ}[G]{\otimes }_{\text{ℤ}}\text{ℤ}[\frac{1}{p}]$ is isomorphic to $K_2(\Gamma ){\otimes }_{\text{ℤ}}\text{ℤ\hspace{0.17em}}[\frac{1}{p}]$ when p ≡ 2, 3, 5, 7. As an application, we give the structure of K₂(ℤ[G]) for G a cyclic p-group or an elementary abelian p-group.

Let (A, 𝔪) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with ℓ(0 :_M I) < ∞, then we show that for a minimal reduction J of I, (0 :_M JI) = (0 :_M I²) if and only if $\ell (0{:}_M{I}^{n+1})=\ell (0{:}_{M}J)\left(\begin{array}{c}n+\text{}d\\ d\end{array}\right)-\ell ((0{:}_{M}J);/(0{:}_{M}I))\left(\begin{array}{c}n+d-1\\ d-1\end{array}\right)$ for all n ≥ 0. Moreover, we study the dual of Burch’s inequality. In particular, the Burch’s inequality becomes an equality if G(I, M) is co-Cohen-Macaulay.

A G-grading on an algebra, where G is an abelian group, is called multiplicity-free if each homogeneous component of the grading is 1-dimensional. We introduce skew root systems of Lie type and skew root systems of Jordan type, and use them to construct multiplicity-free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp., Jordan) algebras are simple. Two families of skew root systems of Lie type (resp., of Jordan type) are constructed and the corresponding Lie (resp., Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.

A subgroup H of G is called ℳ-supplemented in G if there exists a subgroup B of G such that G = HB and H_i B < G for every maximal subgroup H_i of H. In this paper, we use ℳ-supplemented subgroups to study the structure of finite groups and obtain some new characterization about solvability and p-supersolvability for a fixed prime p. Some results in the literature are corollaries of our theorems.

A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a CI-Schur ring. By using this condition we offer short proofs for some known results on the CI-property for decomposable Schur rings over an elementary abelian group of rank at most 5.

We obtain alternative explicit Specht filtrations for the induced and the restricted Specht modules in the Hecke algebra of the symmetric group (defined over the ring A = ℤ[q^1/2, q^−1/2], where q is an indeterminate) using C-bases for these modules. Moreover, we provide a link between a certain C-basis for the induced Specht module and the notion of pairs of partitions.