Algebra Colloquium

In this paper we prove that a finite p-solvable group G is solvable if its every conjugacy class size of p′-elements with prime power order equals either 1 or m for a fixed integer m. In particular, G is 2-nilpotent if 4 does not divide every conjugacy class size of 2′-elements with prime power order.

We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra , introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that is an infinite-dimensional complete Lie algebra, and the universal central extension of in the category of Leibniz algebras is the same as that in the category of Lie algebras.

The Poincaré series of a Coxeter group W measures its growth relative to the generating set R. A well-known formula gives an inductive method of calculating the Poincaré series of W when W is finite. The aim of this paper is to obtain generalizations of this formula.

This paper describes the behaviour of the equimultiple locus of algebroid surfaces under blowing-up for an arbitrary characteristic ground field, extending previous results of the authors for characteristic zero.

In this paper, we study the structure of finite groups G under the assumption that certain abelian subgroups of largest possible exponent of prime power order lie in the 𝔉-hypercenter of G. We improve and extend some results of Ezzat, Shaalan, Elashiry, Asaad, Ramadan, and Buckley.

We study algebras A → B with the property that H₁(A,B,B)=0. They are quite near to smooth algebras, that is why they are called almost smooth algebras. Some applications to the evolution problem of Eisenbud and Mazur are given.

Let (W,S) be a Weyl group of type B_n. In this paper, we find certain nontrivial distinguished involutions in the two-sided cell Ω_t of W with a-value for 1≤ 2t ≤ n and n even (resp., for 1≤ 2t < n and n odd). Besides, for 1≤ i₁< ⋯ < i_t≤ n-1, let L_{i1⋯ it} be a left cell of W with a-value t² and R(L_{i1⋯ it}) ={s_i1,…,s_it}. There are only two involutions y(i₁,i₁)⋯ y(i_t-1,i_t-1) s_it and y(i₁,i₁)⋯ y(i_t,i_t) in the left cell L_{i1⋯ it}. We prove that the former (resp., latter) is a distinguished involution in the left cell L_{i1⋯ it} for t odd (resp., even).

New characteristic properties of left GPGV-rings are given. It is shown that if R is a left GPGV-ring, then for any nonzero element a in R, there is a positive integer n such that aⁿ≠ 0 and (RaR+ l(aⁿ))⊕ L=R for some left ideal L contained in Soc(_RR). As a corollary of this result, we are able to give a positive answer to a question raised by Yue Chi Ming.

We study the normal structure of maximal parabolic subgroups of a Chevalley group over a commutative ring. More precisely, we describe the subgroups of a maximal parabolic subgroup P normalized by the elementary part of its Levi subgroup. As a corollary, we obtain a description of the subgroups in P normalized by its elementary subgroup EP.

Linton in 1978 proved that a λ-large subgroup of an abelian p-group A is totally projective if and only if A is totally projective. In this paper it will be shown that a special λ-large subgroup of a C_Ω-p-group A is summable if and only if A is summable.

Given a commutative Noetherian local ring (R, 𝔪), it is shown that R is Gorenstein if and only if there exists a system of parameters x₁,…,x_d of R which generates an irreducible ideal and

For a given algebra A= 〈A,+,·〉, we can define its anti-symmetric algebra A^-= 〈A^-,+,[ , ]〉 using the commutator [ , ] of A, where the sets A and A^- are the same. We show that there are isomorphic algebras A₁ and A₂ such that their anti-symmetric algebras are not isomorphic. We define a special type Lie algebra and show that it is simple.

In this paper, some methods are given to construct dipolarizations in Kac–Moody algebras as well as their some subalgebras, and an illustrative example is provided. The dipolarizations found in this paper include symmetric and non-symmetric ones.

This paper considers Lie bialgebra structures on Lie algebras of Block type and proves that all such Lie bialgebras are triangular coboundary.

Let R be an associative ring with identity and M an R-module. Let Spec(M) be the set of all prime submodules of M. We topologize Spec(M) with the Zariski topology and prove some useful results.

A finite set of generators for a free product of two groups of type F₃ with a subgroup amalgamated, and an estimation for the upper bound of the second order Dehn functions of the amalgamated free product are carried out.

We consider a generalized superabundant semigroup within the class of semiabundant semigroups, called a supercryptogroup since it is an analogy of a cryptogroup in the class of regular semigroups. We prove that a semigroup S is an -regular supercryptogroup if and only if S can be expressed as a refined semilattice of completely -simple semigroups. Some results on regular cryptogroups are extended to -regular supercryptogroups. Some results on superabundant semigroups are also generalized.