On a minimal set of generators for the invariants of 3 × 3 matrices

Let K be a field of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.     

Free Groups and Subgroups of Finite Index in the Unit Group of an Integral Group Ring

In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-Abelian group G for which every nonlinear irreducible complex representation is of degree 2 and with commutator subgroup G′ a central elementary Abelian 2-group.     

On the Dimensions of Commutative Subalgebras and Subgroups of Nilpotent Algebras and Lie Groups of Class 2

We obtain the functions that bound the dimensions of finite dimensional nilpotent associative or Lie algebras of class 2 over an algebraically closed field in terms of the dimensions of their commutative subalgebras. As a result, we also compute a similar function for complex nilpotent Lie groups of class 2.     

Complete precones on noncommutative integral domains

A subgroup H is called Q-supplemented in a finite group G, if there exists a subgroup K of G such that G = HK and H ∩ K is contained in H _QG , where H _QG is the maximal quasinormal subgroup of G contained in H. In this article, we investigate the influence of Q-supplementation of some primary subgroups in finite groups. Some recent results are generalized.     

Classification of Groups Which Admit a Finite Number of Distinct Right-Orders

Tararin has shown that a non-Abelian group G admits a nonzero finite number of distinct right-orders if and only if G is equipped with a Tararin-type series of some length n. Further, a group which has a Tararin-type series of length n admits 2 ⁿ right-orders. It is known that a group has two right-orders if and only if it is torsionfree Abelian of rank 1. Here we completely classify the groups which admit either four or eight right-orders.     

A Note on Irwin's Conjecture (E.F.I. p-Groups = Thick p-Groups)

We show that in the article of Cutler and Missel (1984 Cutler , D. , Missel , C. ( 1984 ). The structure of C-decomposable p ^ω+n -projective abelian p-groups . Comm. Alg. 12 : 301 – 319 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) there is, in fact, an example of an essentially finitely indecomposable (e.f.i.) abelian p-group which is not thick, and which can be chosen even to be separable, thus answering once again in the negative Irwin's conjecture that the e.f.i. p-groups are precisely the thick ones and that was already resolved in Dugas and Irwin (1992 Dugas , M. , Irwin , J. ( 1992 ). On thickness and decomposability of abelian p-groups . Isr. J. Math. 79 : 153 – 159 . [Google Scholar]).     

弱M-可补子群对合成因子的影响

设G是有限群,P是G的一个Sylow p-子群,1<≤P。考虑|G|的素因子5和7,利用P的每一个阶为|D|的子群H在G中的弱M-可补性质,进一步探究了G的合成因子的结构。     

n-极大子群为共轭可换的有限群

群G的子群H称为G的共轭可换子群,若HHg=HgH,对任意g∈G都成立,本文考查了n-极大子群为共轭可换时对有限群构造的影响.     

On the Normal Subgroup with Exactly Two G-Conjugacy Class Sizes

Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either i or m. Particularly, it is shown that N is Abelian if N ∩ Z（G）=1 and the G-conjugacy class size of every element of N is either 1 or m.