Discussion on the Complex Structure of Nilpotent Lie Groups Gk

Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent Lie group Gk, defined by the nilpotent Lie algebra g/ak, where g is the Lie algebra of G, and ak is an ideal of g. Then, J gives rise to an almost complex structure Jk on Gk. The main conclusion obtained is as follows: if the almost complex structure J of a nilpotent Lie group G is nilpotent, then J can give rise to a left-invariant integrable almost complex structure Jk on the nilpotent Lie group Gk, and Jk is also nilpotent.

Finitely Generated Torsion-free Nilpotent Groups Admitting an Automorphism of Prime Order

Let G be a finitely generated torsion-free nilpotent group and a an automorphism of prime order p of G. If the map ψ ： G → G defined by g^φ = [g, α] is surjective, then the nilpotent class of G is at most h（p）, where h（p） is a function depending only on p. In particular, if α^3 = 1, then the nilpotent class of G is at most 2.

A characteristic condition of finite nilpotent group

This paper gives a characteristic condition of finite nilpotent group under the assumption that all minimal subgroups of G are well-suited in G.

Herz Spaces on Nilpotent Lie Groups and Its Applications

In this paper, we define Herz type spaces on nilpotent Lie groups and give the boundedess of heat kernel and Riesz transform on Herz spaces.

The Jacobson and Brown-McCoy Radicals of Certain Group Rings

In this paper, we determine the Jacobson radicals and Brown-McCoy radicals of group rings of certain non-abelian groups and generalize some known results.

On Semilocal Group Rings

Let R be an associative ring with identity. R is said to be semilocal if R/J（R） is （semisimple） Artinian, where J（R） denotes the Jacobson radical of R. In this paper, we give necessary and sufficient conditions for the group ring RG to be semilocal, where G is a locally finite nilpotent group.

On Weakly Semi-radicable Groups

In this paper, we prove that if a torsion nilpotent group G is a weak semi-radicable group, then every Sylow p-group Gp is a central-by-finite p-group, and moreover Gp's center ζ(GP) satisfies |ζ(GP) : (ζ(GP))P| <∞, that is, ζ(GP) = D×F, where D is a divisible Abelian group, and F is a finite Abelian group.

Two Matrix Theorems Arising from Nilpotent Groups

For a nilpotent group G without π-torsion,and x,y ∈ G,if x^(n)=y^(n) for a T-number n,then x=y;if x^(m)y^(n)=y^(n)x^(m) for n-numbers m,n,then xy=yx.This is a wellknown result in group theory.In this paper,we prove two analogous theorems on matrices,which have independence significance.Specifically,let m be a given positive integer and A a complex square matrix satisfying that(i)all eigenvalues of A are nonnegative,and(i)rank A^(2)=rank A;then A has a unique m-th root X with rank X^(2)=rank X,all eigenvalues of X are nonnegative,and moreover there is a polynomial f(λ)with X=f(A).In addition,let A and B be complex n×n matrices with all eigenvalues nonnegative,and rank A^(2)=rank A,rank B^(2)=rank B;then(i)A=B when A^(r)=B^(r) for some positive integer r,and(i)AB=BA when A^(s)B^(t)=B^(t)A^(s) for two positive integers s and t.

On the Product of Two Nilpotent Subgroups of a Finite Group

It is known that the product of two nilpotent subgroups of a finite group is not necessarily nilpotent.In this paper, we study the influence of the Engel condition on the product of two nilpotent subgroups. Our results generalize some well-known results.

On Coleman Automorphisms of Finite Nilpotent Groups by Cyclic Groups

Let G be a finite group and let N be a nilpotent normal subgroup of G such that G/N is cyclic. It is shown that under some conditions all Coleman automorphisms of G are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.

Subgroup Separability of Certain Generalized Free Products of Nilpotent-by-finite Groups

In this paper, we show that certain generalized free products of nilpotent-by-finite groups are subgroup separable when the amalgamated subgroup is × D where D is in the center of both factors.

Finitely Generated Nilpotent Groups of Infinite Cyclic Commutator Subgroups

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup.Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group offinite rank of infinite cyclic center,we provide a decomposition of G as the product of a generalized extraspecial Z-group and its center.By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial Z-groups,we finally obtain the structure and invariants of the group G.

Isoperimetry of nilpotent groups

This survey gives an overview of the isoperimetric properties of nilpotent groups and Lie groups. It discusses results for Dehn functions and filling functions as well as the techniques used to retrieve them. The content reaches from long standing results up to the most recent development.

A Note on the Heat Kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

In this note,we use Schr?dinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol,which is associated with the rescaled harmonic oscillator.Then we can give an explicit formula for the heat kernel of the rescaled harmonic oscillator for the singularity at the origin.Our results are useful for the general two step nilpotent Lie groups,including the Heisenberg group and H-type group.

Fibonacci Lengths of Certain Nilpotent 2-Groups

In this paper, we study two classes of 2-generated 2-groups of nilpotency class 2 classified by Kluempen in 2002 and also a class of finite 2-groups of high nilpotency class for their Fibonacci lengths. Their involvement in certain interesting sequences of Tribonacci numbers gives us some explicit formulas for the Fibonacci lengths and this adds to the small class of finite groups for which the Fibonacci length are known.

The Weighted Estimates of the Schrodinger Operators on the Nilpotent Lie Group

In this paper we consider the SchrSdinger operator -△G ＋ W on the nilpotent Lie group G where the nonnegative potential W belongs to the reverse H51der class Bq1 for some q1 ≥ D and D is the dimension at infinity of G. The weighted L^p -L6q estimates for the operators W^a（-△G ＋ W）^-β and W^a△G（-△G ＋ W）^-β are obtained.

Determining Sets and Determining Numbers of Finite Groups

A subset D of a group G is a determining set of G if every automorphism of G is uniquely determined by its action on D,and the determining number of G,a(G),is the cardinality of a smallest determining set.A group G is called a DEG-group if α(G)equals(G),the generating number of G.Our main results are as follows.Finite groups with determining number 0 or 1 are classified;finite simple groups and finite nilpotent groups are proved to be DEG-groups;for a given finite group H,there is a DEG-group G such that H is isomorphic to a normal subgroup of G and there is an injective mapping from the set of all finite groups to the set of finite DEG-groups;for any integer k≥2,there exists a group G such that α(G)=2 and(G)≥k.

How does Diagonal Subgroup Embedding Determine the Structure of a Group?

Let G be a finite group.Let D={(g,g)|g∈C},the main diagonalsubgroup of G x G.In this paper,we consider the suitable generalized normalities orindex of D in G x G,some interesting results are obtained.

Finite Groups in Which Every Subgroup Is Abelian or Normal

The main object of this paper is to investigate the finite groups in which every subgroup is either abelian or normal. We obtain a characterization of the groups for the nonnilpotent case, and we also give some properties for the nilpotent case.

Groups Whose Proper Subgroups are Baer Groups

In this paper we classify infinite soluble minimal non-nilpotent-groups, detemine the basic structure of infinite soluble minimal non-Baer-groups, and using famed Heineken-Mohamed- groups we construct an example of minimal non-Baer-group which is not minimal non-nilpotent- group.