Semidirect products OF fuzzy subgroups

The semidirect product of two groups is an important construction in group theory. In this paper we define the semidirect product of fuzzy subgroups and give conditions for it to be a fuzzy subgroup. This extends the work of H. Sherwood on direct products of fuzzy subgroups. We then give an example where a fuzzy subgroup of a semidirect product is a semidirect product of fuzzy subgroups.     

The existence of large commutator subgroups in factors and subgroups of non-nilpotent groups

We prove that any finite non-nilpotent group G must possess certain factors K / M with a large commutator subgroup, where M is nilpotent. These results are related to recent results by M. Herzog and the authors on large commutator subgroups.Received: 26 August 2004     

A characterization of fuzzy subgroups

An ordinary subgroup of a group G is (1) a subset of G, (2) closed under the group operation. In a fuzzy subgroup it is precisely these two notions that lose their deterministic character. A fuzzy subgroup μ of a group (G,·) associates with each group element a number, the larger the number the more certainly that element belongs to the fuzzy subgroup. The closure property is captured by the inequality μ(x · y)?T(μ(x), μ(y)). In A. Rosenfeld's original definition, T was the function ‘minimum’. However, any t-norm T provides a meaningful generalization of the closure property. Two classes of fuzzy subgroups are investigated. The fuzzy subgroups in one class are subgroup generated, those in the other are function generated. Each fuzzy subgroup in these classes satisfies the above inequality with T given by T(a, b) = max(a + b ?1, 0). While the two classes look different, each fuzzy subgroup in either is isomorphic to one in the other. It is shown that a fuzzy subgroup satisfies the above inequality with $\text{T =}\text{‘minimum’}$ if and only if it is subgroup generated of a very special type. Finally, these notions are applied to some abstract pattern recognition problems.     

直积群上几类直觉模糊子群及其投影

给出了直觉模糊子群的性质及其等价命题,并通过定义直觉模糊集的直积与投影,获得了直觉模糊直积群及直觉模糊投影子群,进而在直觉模糊标准子群的条件下,分别讨论了直积群上的直觉模糊正规子群、直觉模糊特征子群、直觉模糊共轭与其直觉模糊投影子群的关系.     

Fuzzy invariant subgroups and fuzzy ideals

The purpose of this paper is to introduce some basic concepts of fuzzy algebra, as fuzzy invariant subgroups, fuzzy ideals, and to prove some fundamental properties. In particular, it will give a characteristic of a field by a fuzzy ideal.     

Groups with few conjugacy classes of non-normal subgroups

The structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.      

Large characteristic subgroups with modular subgroup lattice

It is known that if a group contains an abelian subgroup of finite index, then it also has an abelian characteristic subgroup of finite index. The aim of this paper is to prove that corresponding results hold when abelian subgroups are replaced either by subgroups having a modular subgroup lattice or by quasihamiltonian subgroups.     

Two-knot groups with torsion free abelian normal subgroups of rank two

We show that a torsion free abelian normal subgroup of rank two of a two-knot group which is contained in the commutator subgroup must be free abelian, the centralizer of the abelian subgroup is not contained in the commutator subgroup, and neither of the latter two groups is finitely generated. Furthermore, we characterize algebraically the groups of 2-knots which are cyclic branched covers of twist spun torus knots.     

Groups with finitely many normalizers of non-abelian subgroups

Abstract A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite commutator subgroup. Here the structure of locally graded groups with finitely many normalizers of (infinite) non-abelian subgroups is investigated, and the above result is extended to this more general situation. Keywords: normalizer subgroup, metahamiltonian group Mathematics Subject Classification (2000): 20F24     

Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions

We describe the conjugacy classes of finite subgroups in some split extensions using the notion of 1-cocycle and 1-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank 3 is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field.     

On the coarse-geometric detection of subgroups

We generalize [20] to give sufficient conditions, primarily on coarse geometry, to ensure that a subset of a Cayley graph is a finite Hausdorff distance from a subgroup. Using this result, we prove a partial converse to the Flat Torus Theorem for CAT(0) groups. Also using this result, we give sufficient conditions for subgroups and splittings to be invariant under quasi-isometries.     

Separable subgroups of mapping class groups

We investigate separability questions for the mapping class group of a surface. While this group is not subgroup separable in general, we prove a large family of interesting subgroups are separable. This includes many classically studied subgroups such as solvable subgroups, Heegaard and Handlebody groups, geometric subgroups, and all the terms in the Johnson filtration.     

Borel subgroups of Polish groups

We study three classes of subgroups of Polish groups: Borel subgroups, Polishable subgroups, and maximal divisible subgroups. The membership of a subgroup in each of these classes allows one to assign to it a rank, that is, a countable ordinal, measuring in a natural way complexity of the subgroup. We prove theorems comparing these three ranks and construct subgroups with prescribed ranks. In particular, answering a question of Mauldin, we establish the existence of Borel subgroups which are -complete, α?3, and -complete, α?2, in each uncountable Polish group. Also, for every α<ω₁ we construct an Abelian, locally compact, second countable group which is densely divisible and of Ulm length α+1. All previously known such groups had Ulm length 0 or 1.     

Stacky abelianization of algebraic groups

We prove a conjecture of Drinfeld regarding restriction of central extensions of an algebraic group G to the commutator subgroup. As an application, we construct the “true commutator” of G. The quotient of G by the action of the true commutator is the universal commutative group stack to which G maps.     

Finite A-groups with complementable nonmetacyclic subgroups

We study groups G that satisfy the following conditions (i) G is a finite solvable group with nonidentity primary metacyclic second commutator subgroup (ii) all Sylow subgroups of G are elementary Abelian. We describe the structure of groups of this type with complementable nonmetacyclic subgroups. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 607–615, May, 2006.     

Finite index subgroups of graph products

We prove that every quasiconvex subgroup of a right-angled Coxeter group is an intersection of finite index subgroups. From this we deduce similar separability results for other types of groups, including graph products of finite groups and right-angled Artin groups.      

A construction of all normal subgroup lattices of 2-transitive automorphism groups of linearly ordered sets

We give a complete classification and construction of all normal subgroup lattices of 2-transitive automorphism groupsA(Ω) of linearly ordered sets (Ω, ≦). We also show that in each of these normal subgroup lattices, the partially ordered subset of all those elements which are finitely generated as normal subgroups forms a lattice which is closed under even countably-infinite intersections, and we derive several further group-theoretical consequences from our classification. This research was supported by an award from the Minerva-Stiftung, München. The work was done during a stay of the first-named author at The Hebrew University of Jerusalem in fall 1982. He would like to thank his colleagues in Jerusalem for their hospitality and a wonderful time.     

On intersections of solvable Hall subgroups in finite nonsolvable groups

The author continues the investigation of intersections of Hall subgroups in finite groups. Previously, the author proved that in the case when a Hall subgroup is Sylow there are three subgroups conjugate to it such that their intersection coincides with the maximal normal primary subgroup. A similar assertion holds for Hall subgroups in solvable groups. The aim of this paper is to construct examples of a (nonsolvable) group in which the intersection of any four subgroups conjugate to some Hall subgroup is nontrivial.     

Non-nudgable subgroups of permutations

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is called non-nudgable if the cardinality of this set always remains the same when replacing the initial permutation with its inverse. It is called nudgable otherwise. We show that all full permutation groups, standard dihedral groups, half of the alternating groups, and any abelian subgroup are non-nudgable. In the right probabilistic sense, it is thus quite likely that a randomly generated subgroup is non-nudgable. However, the other half of the alternating groups are nudgable. We also construct a smallest possible nudgable group, a 6-element subgroup of the permutation group on 4 elements.