Algebra Colloquium

In this paper, we characterize the finite solvable groups with non-complete character degree graphs by proving the following theorem, which generalizes a conjecture by Huppert.

Suppose that G is a finite solvable group and p is a prime number dividing the degree of some irreducible character of G. If there is another such prime number q such that pq does not divide the degree of any irreducible character of G, then both p-length ℓ_p(G) and q-length ℓ_q(G) of G are at most two, and ℓ_p(G)+ ℓ_q(G)=4 if and only if pq=6 with Q^G/Z_φ(Q^G)≅ 3²:GL(2,3), where Q_G is generated by all Sylow 2-subgroups of G and Z_φ(G) is a normal nilpotent subgroup of G. Moreover, the bounds are best possible.

It is proved that a finite group G=AB is supersoluble provided that A and B are such supersolube subgroups of G that every cyclic subgroup of A permutes with every Sylow subgroup of B and if in return every cyclic subgroup of B permutes with every Sylow subgroup of A.

Let G be a group and let φ (G) be the least integer k such that G^(k)=G^(k+1). If no such k exists, then φ (G)=∞ and we write G ∈ 𝓤. We are interested in the questions which Coxeter groups are in 𝓤 and how large a finite number φ (G) can be for Coxeter groups G. In this paper, we give full answers in the 3- and 4-generator cases.

Table Algebras (A,B) with distinct non-real basis elements a, b∈ B satisfying , (m, n∈ ℝ⁺) are studied in this paper. The two cases where either a is real and b is non-real or a, b are both real were completely classified by Arad and Blau. Here, we study the remaining case where a and b are non-real.

Let n be a positive integer. A ring R is called n-clean if every element of R can be written as a sum of an idempotent and n units in R. The class of n-clean rings contains clean rings and (S,n)-rings (i.e., every element is a sum of no more than n units). In this paper, we investigate some properties on n-clean rings. There exists a clean and (S,3)-ring which is not an (S,2)-ring. If R is a ring satisfying (SI), then the polynomial ring R[x] is not n-clean for any positive integer n. An example shows that for any positive integer n> 1, there exists a non n-clean ring R such that the 2× 2 matrix ring M₂(R) over R is n-clean.

In this paper, we study lattices of subgroups that can be used to obtain lifts of π-partial characters. In particular, we find a condition on the lattices so that the associated lifts will be well-behaved regarding induction and restriction on normal subgroups. We show that the lattices obtained from all subnormal subgroups have this property, but the lattices obtained from all normal subgroups do not.

An R-module M is said to have the cancellation property provided that M⊕ B ≅ M⊕ C implies B ≅ C for any pair of R-modules B and C. We obtain a characterization of the cancellation property for projective R-modules. With this result, it is proved that Dedekind domains have the cancellation property; and if R is a Prüfer domain, then R⊕ B ≅ R⊕ C implies B ≅ C for any pair of finitely generated R-modules B and C.

The theory of hyperidentities and hypervarieties is based on the fact that the set Hyp(τ) of all hypersubstitutions of a fixed type τ forms a monoid, with a Galois connection between submonoids of this monoid and complete sublattices of the lattice of all varieties of type τ. For this reason, there is interest in studying the semigroup or monoid properties of Hyp(τ) and its submonoids. One approach is to study the five relations known as Green's relations definable on any semigroup. In this paper, we consider the type τ = (n) with one n-ary operation symbol for n≥ 1, and the submonoid Reg(n) of regular hypersubstitutions. We characterize Green's relations on every subsemigroup of Reg(n); then using this characterization we describe which subsemigroups of Reg(n) are 𝒢-subsemigroups of Reg(n) defined by Levi.

This paper discusses symmetry operators of the tensor product of free modules of finite rank over domains, shows two formulas to calculate the rank of their images, and applies them to the modular symmetry classes of tensors.

In this paper, we show that the simple group S₈(2) is irrecognizable by its spectrum.

The notion of δ-semiregular modules is introduced in terms of δ-small submodules. It is shown that a ring R is a δ-semiregular ring if and only if R_R is a δ-semiregular module, if and only if _RR is a δ-semiregular module, if and only if every finitely presented module has a direct-projective δ-cover.

For a symmetrizable Borcherds–Cartan matrix A with integer entries and even diagonal entries, we show that there exists a k-species 𝓢 over the finite field k such that 𝓢 and the Borcherds–Cartan matrix provide the same bilinear form. We also show that the number of isomorphism classes of indecomposable representations of any valued graph with fixed dimension vector is a polynomial, and is independent of the orientation of the valued graph. This extends to the situation of valued graphs with loops.

Let I be an exchange QB-ideal of a unital ring R. It is shown that for any regular A∈ M_n(I), there exist U, V∈ M_n(R)_q^-1∩ (I_n+M_n(I)) such that UAV= diag(e₁,…,e_n) for some idempotents e₁,…,e_n ∈ I.

An algebra is called term-linear if every term is equal to a linear term or a constant and every n-ary linear term is essential. The term-linearity is a generalization of the linearity of polynomials over ordinary linear algebras or vector spaces. In this paper, as a sequel to a previous paper, we present an explicit normal form for terms over an affine space over GF(3) by binary trees, which shows that the word problem for affine spaces over GF(3) is solvable. Using this normal form, we count the number of all essentially n-ary linear terms and apply the Csákány formula to show that an affine space over GF(3) is term-linear.

Let A and B be prime rings and θ: A→ B a bijective additive map such that θ(x) θ(y)=0 for all x, y∈ A with xy=0. Suppose that the maximal right quotient ring Q(A) of A contains a nontrivial idempotent e such that eA ∪ Ae ⊆ A. Then there exists λ ∈ C(B), the extended centroid of B, such that θ(xy) = λ θ(x) θ(y) for all x, y ∈ A. This removes the assumption deg(B)≥ 3 in the recent theorem due to Chebotar, Ke and Lee.

Let e ≥ 1 be an integer and S={x₁,…,x_n} a set of n distinct positive integers. The matrix ([x_i, x_j]^e) having the power [x_i, x_j]^e of the least common multiple of x_i and x_j as its (i, j)-entry is called the power least common multiple (LCM) matrix defined on S. The set S is called gcd-closed if (x_i,x_j) ∈ S for 1≤ i, j≤ n. Hong in 2004 showed that if the set S is gcd-closed such that every element of S has at most two distinct prime factors, then the power LCM matrix on S is nonsingular. In this paper, we use Hong's method developed in his previous papers to consider the next case. We prove that if every element of an arbitrary gcd-closed set S is of the form pqr, or p²qr, or p³qr, where p, q and r are distinct primes, then except for the case e=1 and 270, 520 ∈ S, the power LCM matrix on S is nonsingular. We also show that if S is a gcd-closed set satisfying x_i< 180 for all 1≤ i≤ n, then the power LCM matrix on S is nonsingular. This proves that 180 is the least primitive singular number. For the lcm-closed case, we establish similar results.

Suppose G is a finite group and D is a normal p-subgroup of G with |D|=p^d. Let G_P denote the set of p-elements of G and Φ(g)={(a,b) ∈ G_P× G_P | ab=g}. We show that G has a p-block with D as a defect group if and only if there exists a p-regular element g of G with D as a Sylow p-subgroup of C_G(g) such that |Φ(g)| ≠ 0 (mod p^d+1).

The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, alled the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.