Groups Whose Finite Homomorphic Images are Metahamiltonian

A group G is metahamiltonian if all its non-abelian subgroups are normal. It is proved here that a finitely generated soluble group is metahamiltonian if and only if all its finite homomorphic images are metahamiltonian; the behaviour of soluble minimax groups with metahamiltonian finite homomorphic images is also investigated. Moreover, groups satisfying the minimal condition on non-metahamiltonian subgroups are described.     

Finite Homomorphic Images of Groups of Finite Rank

Let π be a finite set of primes. We prove that each soluble group of finite rank contains a finite index subgroup whose every finite homomorphic π-image is nilpotent. A similar assertion is proved for a finitely generated group of finite rank. These statements are obtained as a consequence of the following result of the article: Each soluble pro-π-group of finite rank has an open normal pronilpotent subgroup.

     

Uncountable Cofinalities of Permutation Groups

A sufficient criterion is found for certain permutation groupsG to have uncountable strong cofinality, that is, they cannotbe expressed as the union of a countable, ascending chain (H_i)_i     

MS代数的极大同态象

该文找到了MS代数的商代数分别为De Morgan代数、Boole代数及Stone代数的最小同余关系,并借助MS代数的对偶理论,得到了MS代数的极大同态象的对偶表示.     

模糊粗糙群中的同态映射(英文)

This paper is devoted to the discussion of homomorphic properties of fuzzy rough groups.The fuzzy approximation space was generated by fuzzy normal subgroups and the fuzzy rough approximation operators were discussed in the frame of fuzzy rough set model.The basic properties of fuzzy rough approximation operators were obtained.     

不充当自同构群的有限群

设ｐ为奇素数，本文将用一些新的技巧来证明，当Ｐ是阶小于Ｐ＾１１的交换Ｐ－群时，自同构群方程Ａｕｔ（Ｘ）＝Ｐ无解。这个结果使ＭａｃｈＨａｌｅ在１９８３年的工作得到了突破，并且我们所给的方法具有广泛性。     

Sol Geometry Groups are not Asynchronously Automatic

Lattices in the 3-dimensional Lie group Sol are shown to benot asynchronously automatic. The proof consists of a combinationof geometry and logical complexity arguments. 2000 MathematicsSubject Classification: 20F10, 20F16.     

Uncountable products of determined groups need not be determined

If H is a dense subgroup of G, we say that H determines G if their groups of characters are topologically isomorphic when equipped with the compact open topology. If every dense subgroup of G determines G, then we say that G is determined. The importance of this property is justified by the recent generalizations of Pontryagin-van Kampen duality to wider classes of topological Abelian groups. Among other results, we show (a) _⊕i∈IR determines the product _∏i∈IR if and only if I is countable, (b) a compact group is determined if and only if its weight is countable. These answer questions of Comfort, Raczkowski and the third listed author. Generalizations of the above results are also given.     

Groups whose elements are not conjugate to their powers

We call a finite group irrational if none of its elements is conjugate to a distinct power of itself. We prove that those groups are solvable and describe certain classes of these groups, where the above property is only required for p-elements, for p from a prescribed set of primes.     

Solvable Baumslag–Solitar Groups are not Almost Convex

The arguments of Cannon, Floyd, Grayson and Thurston (Geom. Dedicata (1989), 291–300) showing that solvegeometry groups are not almost convex apply to solvable Baumslag-Solitar groups.     

Finitary Power Semigroups of Infinite Groups are not Finitely Generated

For a semigroup S, the finitary power semigroup of S, denotedP_f(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that P_f(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary).     

Groups of p-absolute Galois type that are not absolute Galois groups

Let p be a prime. We study pro-p groups of p-absolute Galois type, as defined by Lam–Liu–Sharifi–Wake–Wang. We prove that the pro-p completion of the right-angled Artin group associated to a chordal simplicial graph is of p-absolute Galois type, and moreover it satisfies a strong version of the Massey vanishing property. Also, we prove that Demushkin groups are of p-absolute Galois type, and that the free pro-p product — and, under certain conditions, the direct product — of two pro-p groups of p-absolute Galois type satisfying the Massey vanishing property, is again a pro-p group of p-absolute Galois type satisfying the Massey vanishing property. Consequently, there is a plethora of pro-p groups of p-absolute Galois type satisfying the Massey vanishing property that do not occur as absolute Galois groups.     

Groups of even type which are not of even characteristic,I

In the ongoing revision of the classification of the finite simple groups there is a subdivision into two classes of groups, which reflects whether semisimple elements or unipotent elements are the primary focus of the investigation. While semisimple methods naturally lead to the definition of groups of even type, unipotent methods, notably the amalgam method, naturally lead to groups of even characteristic. This paper clarifies the relationship between the two definitions and thus makes the amalgam method available for use in the classification of groups of even type.     

Groups with subnormality for all subgroups that are not finitely generated

Summary We characterize the groups given in the title in the case of locally finite, locally nilpotent and radical groups.

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Sunto I gruppi con tutti i sottogruppi nonfinitamente generati subnormali sono caratterizzati nella classe dei gruppi localmente finiti, localmente nilpotenti e radicali.

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This work was carried out while the second author visited the University of Würzburg with the aid of Deutsche Forschungsgemeinschaft (DFG). The second author is grateful to DFG for making this visit possible and to Mathematisches Institut Würzburg for the hospitality he enjoyed.     

Subalgebras and Homomorphic Images of Algebras Having the CEP and the WCIP

In the present paper we consider algebras satisfying both the congruence extension property (briefly the CEP) and the weak congruence intersection property (WCIP for short). We prove that subalgebras of such algebras have these properties. We deduce that a lattice has the CEP and the WCIP if and only if it is a two-element chain. We also show that the class of all congruence modular algebras with the WCIP is closed under the formation of homomorphic images.     

All Groups are Outer Automorphism Groups of Simple Groups

It is shown that each group is the outer automorphism groupof a simple group. Surprisingly, the proof is mainly based onthe theory of ordered or relational structures and their symmetrygroups. By a recent result of Droste and Shelah, any group isthe outer automorphism group Out (Aut T) of the automorphismgroup Aut T of a doubly homogeneous chain (T, ). However, AutT is never simple. Following recent investigations on automorphismgroups of circles, it is possible to turn (T, ) into a circleC such that Out (Aut T) Out (Aut C). The unavoidable normalsubgroups in Aut T evaporate in Aut C, which is now simple,and the result follows.