Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

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     

Some explicit bounds in groups with a finite number of non-normal subgroups

The number of conjugacy classes of non-normal subgroups is an invariant of a group G denoted by v{G). In this paper explicit upper bounds for the order of the commutator subgroup G' and for the index of the centre in G, [G : Z(G)] are given for a group G with only a finite number of non-normal subgroups. The bounds provided are functions of v(G) and the primes appearing as orders of elements of G.     

On finite groups with few non-normal subgroups

In this paper we characterize the finite groups having exactly two conjugacy classes of non-normal subgroups.     

Groups with many subgroups having a transitive normality relation

A group is said to be aT-group if all its subnormal subgroups are normal. The structure of groups satisfying the minimal condition on subgroups that do not have the propertyT is investigated. Moreover, locally soluble groups with finitely many conjugacy classes of subgroups which are notT-groups are characterized.     

Groups with finiteness conditions on the lower central series of non-normal subgroups

It is known that any locally graded group with finitely many derived subgroups of non-normal subgroups is finite-by-abelian. This result is generalized here, by proving that in a locally graded group G the subgroup \(\gamma _{k}(G)\) is finite if the set \(\{\gamma _{k}(H)\;|\;H\le G,\,H\ntriangleleft G\}\) is finite. Moreover, locally graded groups with finitely many kth terms of lower central series of infinite non-normal subgroups are also completely described.     

Morse theory and conjugacy classes of finite subgroups

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F _n inside mapping class groups, Aut(), and Out() which have infinitely many conjugacy classes of finite-order elements.      

Finiteness properties for subgroups of GL (n,\mathbb Z)(n,\mathbb Z)

Abstract. We construct finitely presented subgroups of GL that have infinitely many conjugacy classes of finite subgroups. This answers a question of Grunewald and Platonov. We suggest a variation on their question. Received: 26 August 1999 / Revised: 28 September 1999 / Published online: 8 May 2000     

A combination theorem for Veech subgroups of the mapping class group

In this paper we prove a combination theorem for Veech subgroups of the mapping class group analogous to the first Klein–Maskit combination theorem for Kleinian groups in which two Fuchsian subgroups are amalgamated along a parabolic subgroup. As a corollary, we construct subgroups of the mapping class group (for all genera at least 2), which are isomorphic to non-abelian closed surface groups in which all but one conjugacy class (up to powers) is pseudo-Anosov. Received: October 2004 Revision: April 2005 Accepted: April 2005 C.J.L.’s work was partially supported by an NSF postdoctoral fellowship. A.W.R’s work was partially supported by an NSF grant.     

New properties of lattices in Lie groups

We study finite extension groups of lattices in Lie groups which have finitely many connected components. We show that every non-cocompact Fuchsian group (these are the non-cocompact lattices in $\text{PSL}\text{(2,}\text{R}\text{)}$) has an extension group of finite index which is not isomorphic to a lattice in a Lie group with finitely many connected components. On the other hand we prove that these are, in an appropriate sense, the only lattices in Lie groups which have extension groups of this kind. We also show that an extension group of finite index of a lattice in a Lie group with finitely many connected components has only finitely many conjugacy classes of finite subgroups. To cite this article: F. Grunewald, V. Platonov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Fibre products, non-positive curvature, and decision problems

We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of . Received: October 7, 1999.     

Atoroidal Surface Bundles Over Surfaces

A surface-by-surface group is an extension of a non-trivial orientable closed surface group by another such group. It is an open question as to whether every such group contains a free abelian subgroup of rank 2. We show that, for given base and fibre genera, all but finitely many isomorphism classes of surface-by-surface group contain such an abelian subgroup. This can be rephrased in terms of atoroidal surface bundles over surfaces, or in terms of purely loxodromic surface subgroups of the mapping class groups.     

Finite Groups with Few Non-Abelian Subgroups

Let G be a finite group and τ(G) denote the number of conjugacy classes of all non-abelian subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In this paper, finite groups with τ(G) ≤ |π(G)| are classified completely. Furthermore, finite nonsolvable groups with τ(G) = |π(G)| +1 are determined.     

A note on groups whose non-normal subgroups are either abelian or minimal non-abelian

A group G is called parahamiltonian if each non-normal subgroup of G is either abelian or minimal non-abelian. Thus all biminimal non-abelian groups are parahamiltonian, and the class of parahamiltonian groups contains the important class of metahamiltonain groups, introduced by Romalis and Sesekin about 50 years ago. The aim of this paper is to describe the structure of locally graded parahamiltonian groups.

     

On Conjugacy p-Separability of Free Products of Two Groups with Amalgamation

It is proved that a free product of two finite p-groups with amalgamated central subgroups is a conjugacy p-separable group. With the help of this result, it is proved that a free product with amalgamated subgroups of two finitely generated Abelian groups is a residually finite p-group if and only if it is conjugacy p-separable.     

Classification of finite groups according to the number of conjugacy classes

We consider the problem of the classification of finite groups according to the number of conjugacy classes through the classification of all the finite groups with many minimal normal subgroups.     

A New Characterization of the Mathieu Groups and Alternating Groups with Degrees Primes

In the past thirty years, several kinds of quantitative characterizations of finite groups especially finite simple groups have been investigated by many mathematicians. Such as quantitative characterizations by group order and element orders, by element orders alone, by the set of sizes of conjugacy classes, by dimensions of irreducible characters, by the set of orders of maximal abelian subgroups and so on. Here the authors continue this topic in a new area tending to characterize finite simple groups with given orders by some special conjugacy class sizes, such as largest conjugacy class sizes, smallest conjugacy class sizes greater than 1 and so on.     

On decomposability of finite groups

Let G be a finite group. A normal subgroup N of G is a union of several G-conjugacy classes, and it is called n-decomposable in G if it is a union of n distinct G-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.     

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.     

Groups whose non-normal subgroups have a transitive normality relation

A group is calledmetahamiltonian if all its non-normal subgroups are abelian. The structure of metahamiltonian groups has been investigated by Romalis and Sesekin. In this paper groups are studied in which every non-normal subgroup has a transitive normality relation.