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.

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.

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.

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.

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.

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.

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.

Sensitive Group Actions on Regular Curves of Almost≤n Order

Let X be a regular curve and n be a positive integer such that for every nonempty open set U⊂X,there is a nonempty connected open set V⊂U with the cardinality|∂_(X)(V)|≤n.We show that if X admits a sensitive action of a group G,then G contains a free subsemigroup and the action has positive geometric entropy.As a corollary,X admits no sensitive nilpotent group action.

On the Residual Finiteness of a Class of Infinite Soluble Groups

Let A be a completely decomposable homogeneous torsion-free abelian group of rank n(n≥2).Let m(n)=A×(a)be the split extension of A by an automorphismαwhich is a cyclic permutation of the direct components twisted by a rational integer m.Then Om(n)is an infinite soluble group.In this paper,the residual finiteness of Om(n)is investigated.

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.

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.

On Local Nilpotency of the Normal Subgroups of a Group

A group G is said to have property μ whenever N is a non-locally nitpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property v satisfy property μ, where a group G is said to have property v if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP（G） is the Hirsch-Plotkin radical of G, and φf （G） is the intersection of all the maximal subgroups of finite index in G （here φf（G） = G if no such maximal subgroups exist）. It is shown that a group G has property μ if and only if HP（G/φf（G）） = HP（G）/φf（G） and that the class of groups with property v is a proper subclass of that of groups with property it. Also, the structure of the normal subgroups of a group： nilpotency with the minimal condition, is investigated.

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.

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.

On the Unitriangular Groups over Rational Numbers Field

Let U(n,Q)be the group of all n×n(upper)unitriangular matrices over rational numbers field Q.Let S be a subset of U(n,Q).In this paper,we prove that S is a subgroup of U(n,Q)if and only if the(i,j)-th entry S;satisfies some condition(see Theorem 3.5).Furthermore,we compute the upper central series and the lower central series for S,and obtain the condition that the upper central series and the lower central series of S coincide.

The Structure of a Finite Group Which is the Product of Two Subgroups with Some Subnormal Subgroups

Let G be a group and G=G_(1)G_(2) where G_(i) are subgroups of G.In this paper,we investigate the structure of G under the conditions that some subgroups of G_(i) are subnormal in G.

The Influence of s-Conditional Permutability of Subgroups on the Structure of Finite Groups

Let G be a finite group. Fix a prime divisor p of IGI and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Ma（P） denote a family of maximal subgroups P1, P2 , Pd of P satisfying ∩^di=1 Pi = Ф（P）, the Frattini subgroup of P. In this paper, we shall investigate the influence of s-conditional permutability of the members of some fixed .Md（P） on the structure of finite groups. Some new results are obtained and some known results are generalized.