Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable

Thompson’s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson’s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.     

Congruence extension,Hamiltonian and Abelian properties in locally finite varieties

We prove that in a locally finite variety with the congruence extension property, locally solvable congruences are central and locally solvable algebras are Hamiltonian. Also, we prove that a maximal subuniverse of a finite algebra in an Abelian variety is identical with an equivalence class of some congruence.Presented by H. P. Gumm.     

A rigidity theorem for automorphism groups of trees

A group G is called unsplittable if Hom(G, ℤ) = 0 and this group is not a non-trivial amalgam. Let X be a tree with a countable number of edges incident at each vertex and G be its automorphism group. In this paper we prove that the vertex stabilizers are unsplittable groups. Bass and Lubotzky proved (see [3]) that for certain locally finite trees X, the automorphism group determines the tree X (that is, knowing the automorphism group we can “construct” the tree X). We generalize this Theorem of Bass and Lubotzky, using the above result. In particular we show that the Theorem holds even for trees which are not locally finite. Moreover, we prove that the permutation group of an infinite countable set is unsplittable and the infinite (or finite) cartesian product of unsplittable groups is an unsplittable group as well. This research was supported by the European Social Fund and National resources-EPEAEK II grant Pythagoras 70/3/7298.     

On Some Automorphisms of Orthogonal Groups in Odd Characteristic

In this paper, it is proved that the simple orthogonal groups O _2n+1(q) and O _2n ^± (q) (where q is odd) cannot be automorphism groups of finite left distributive quasigroups. This is a particular case of the conjecture stating that the automorphism group of a left distributive quasigroup is solvable. To complete the proof of the conjecture, one must test all finite groups.     

以幂零李代数R_n为幂零根基的可解李代数(英文)

In this paper we explicitly determine automorphism group of filiform Lie algebra Rn to find the indecomposable solvable Lie algebras with filiform Lie algebra Rn nilradicals.We also prove that the indecomposable solvable Lie algebras with filiform Rn nilradicals is complete.     

On anticonformal automorphisms of Riemann surfaces with nonembeddable square

In this paper we present an example of an anticonformal automorphism whose square has prime order and is not embeddable. We prove that every embeddable automorphism of odd order of a compact Riemann surface is the square of an orientation-reversing self-homeomorphism. Finally we study whether a conformal involution is the square of an orientation-reversing automorphism.

     

Rigidity of group actions on solvable Lie groups

We establish analogs of the three Bieberbach theorems for a lattice in a semidirect product where is a connected, simply connected solvable Lie group and is a compact subgroup of its automorphism group. We first prove that the action of on is metrically equivalent to an action of on a supersolvable Lie group. The latter is shown to be determined by itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group as a lattice in a semidirect product with being a finite group whose order is bounded by a constant only depending on the dimension of . This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup that the quotient is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group. Received: 27 August 1999     

非幂零极大子群共轭类类数给定的有限群

证明了非幂零极大子群共轭类类数等于2的有限群必可解,并给出了非幂零极大子群同阶类类数等于2的非可解群的等价刻画.     

The finiteness of the number of symmetrical 2-extensions of the <Emphasis Type="Italic">d</Emphasis>-dimensional grid and similar graphs

We prove that there are only finitely many symmetrical 2-extensions of a locally finite graph whenever the automorphism group of the graph has an abelian subgroup of finite index (this case is of interest for certain applications). Some refinements and generalizations of this result are also obtained.     

The <Emphasis Type="Italic">F</Emphasis>-local subgroups of the groups with a generalized finite element of order 2 or 4

We prove the existence of infinite subgroups with nontrivial locally finite radicals and of infinite locally finite subgroups in the groups with almost finite almost solvable elements of order 2 and 4 and in the groups with almost finite elements.     

一类不能作为自同构群的奇阶群

本文考虑如下问题：怎样的有限群可以作为另一个有限群的全自同构群？我们首先证明，若有限群Ｋ有一个正规Ｓｙｌｏｗｐ－子群使得｜Ｋ：Ｚ（Ｋ）｜ｐ＝ｐ２，那么Ｋ有２阶自同构．利用这个结果，我们证明了，若奇阶群Ｇ具有阶Ｐｓｍ（１≤ｓ≤３），ｐ为｜Ｇ｜的最小素因子，ｐｍ，ｍ无立方因子，则Ｇ不可能作为全自同构群．     

Generalization of fischer's theorem for distributive quasigroups

We generalize Fischer's well-known theorem which asserts that the right associated group of a finite distributive quasigroup is solvable. We prove that the associated group of a finitely generated distributive quasigroup is solvable and that the associated group of any distributive quasigroup is locally solvable.Translated from Matematicheskie Zametki, Vol. 23, No. 4, pp. 521–526, April, 1978.     

Generalized quadrangles admitting a sharply transitive Heisenberg group

All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.      

Coleman自同构群的投射极限

在这篇注记中,利用群的投射极限性质给出了有限可解群的Coleman自同构群的一个具体构造.作为应用,证明了二面体群的Coleman外自同构群或者是1或者是一个初等阿贝尔2-群.     

Coleman自同构群的投射极限

在这篇注记中，利用群的投射极限性质给出了有限可解群的Coleman自同构群的一个具体构造.作为应用，证明了二面体群的Coleman外自同构群或者是1或者是一个初等阿贝尔2-群.     

有限素数度弧正则图

如果一个图的全自同构群在其弧集上正则,则称此图为弧正则图.本文刻画素数度的立方自由阶弧正则图,证明任何素数度2倍奇立方自由阶弧正则图都是正规或二部正规Cayley图,且不存在任意素数度4倍奇立方自由阶的弧正则图,推广了一些已知的结果,得到阶为8倍奇平方自由阶素数度弧正则图的分类,并发现新的弧正则图类.此外,基于所得的结果,我们提出一个猜想和有待后续研究的一些问题.     

A generalization of Schmidt’s Theorem on groups with all subgroups nilpotent

According to Schmidt’s Theorem a finite group whose proper subgroups are all nilpotent (or a finite group without non-nilpotent proper subgroups) is solvable. In this paper we prove that every finite group with less than 22 non-nilpotent subgroups is solvable and that this estimate is sharp.     

On the Shunkov groups acting freely on Abelian groups

We prove the existence of a locally finite periodic part in every rank 1 Shunkov group with solvable finite subgroups, in every Shunkov group acting freely on an abelian group, and in the groups of affine transformations of neardomains with finite elements.     

Odd symplectic flag manifolds

We define the odd symplectic Grassmannians and flag manifolds, which are smooth projective varieties equipped with an action of the odd symplectic group, analogous to the usual symplectic Grassmannians and flag manifolds. Contrary to the latter, which are the flag manifolds of the symplectic group, the varieties we introduce are not homogeneous. We argue nevertheless that in many respects the odd symplectic Grassmannians and flag manifolds behave like homogeneous varieties; in support of this claim, we compute the automorphism group of the odd symplectic Grassmannians and we prove a Borel-Weil-type theorem for the odd symplectic group.