New Classes Of Groups Research Articles

Abelian invariants and a reduction theorem for the modular isomorphism problem

We show that elementary abelian direct factors can be disregarded in the study of the modular isomorphism problem. Moreover, we obtain four new series of abelian invariants of the group base in the modular group algebra of a finite p-group. Finally, we apply our results to new classes of groups.

W⁎ and C⁎-superrigidity results for coinduced groups

In this paper we explore a generic notion of superrigidity for von Neumann algebras L(G) and reduced C⁎-algebras Cr⁎(G) associated with countable discrete groups G. This allows us to classify these algebras for various new classes of groups G from the realm of coinduced groups.

Groups whose derived subgroup is not supplemented by any proper subgroup

In this paper, we introduce two new classes of groups that are described as weakly nilpotent and weakly solvable groups. A group $G$ is weakly nilpotent if its derived subgroup does not have a supplement except $G$ and a group $G$ is weakly solvable if its derived subgroup does not have a normal supplement except $G$. We present some examples and counter-examples for these groups and characterize a finitely generated weakly nilpotent group. Moreover, we characterize the nilpotent and solvable groups in terms of weakly nilpotent and weakly solvable groups. Finally, we prove that if $F$ is a free group of rank $n$ such that every normal subgroup of $F$ has rank $n$, then $F$ is weakly solvable.

Triple transitivity and non‐free actions in dimension one

The transitivity degree of a group G $G$ is the supremum of all integers k $k$ such that G $G$ admits a faithful k $k$ -transitive action. Few obstructions are known to impose an upper bound on the transitivity degree for infinite groups. The results of this article provide two new classes of groups whose transitivity degree can be computed, as a corollary of a classification of all 3-transitive actions of these groups. More precisely, suppose that G $G$ is a subgroup of the homeomorphism group of the circle Homeo ( S 1 ) $\operatorname{Homeo}(\mathbb {S}^1)$ or the automorphism group of a tree Aut ( T ) $\operatorname{Aut}(\mathcal {T})$ . Under natural assumptions on the stabilizers of the action of G $G$ on S 1 $\mathbb {S}^1$ or ∂ τ $\partial\tau$ , we use the dynamics of this action to show that every faithful action of G $G$ on a set that is at least 3-transitive must be conjugate to the action of G $G$ on one of its orbits in S 1 $\mathbb {S}^1$ or ∂ T $\partial \mathcal {T}$ .

Topological groups without infinite precompact continuous homomorphic images

Let P be a property of topological groups. We say that a topological group G is MinAP moduloP if its image f[G] under each continuous homomorphism f:G→K from G to a compact group K has property P. When P is the property of being the trivial group, MinAP modulo P groups are precisely the minimally almost periodic groups of von Neumann and Wigner. We give a characterization of these new classes of groups for five properties P: finite, bounded, torsion, compact and connected. We show that these five classes are distinct and differ from the standard class of minimally almost periodic groups. We equip every Abelian group with a Hausdorff MinAP modulo finite group topology, thereby showing that, unlike the classical minimal almost periodicity, the property MinAP modulo finite imposes no restrictions whatsoever on the algebraic structure of the underlying group. At last but not least, we prove that the quotient group G/n(G) of every topological group G with respect to its von Neumann kernel n(G) is the categorical reflection of G in the class of maximally almost periodic (MAP) groups, and the natural quotient map plays the role of reflection homomorphism. As a consequence, this quotient group (G/n(G))+ equipped with its Bohr topology is topologically isomorphic to the image of G under the canonical homomorphism to its Bohr compactification.

Inverted orbits of exclusion processes, diffuse-extensive-amenability, and (non-?)amenability of the interval exchanges

The recent breakthrough works [9, 11, 12] which established the amenability for new classes of groups, lead to the following question: is the action W(\mathbb Z^d) \curvearrowright \mathbb Z^d extensively amenable? (Where W(\mathbb Z^d) is the wobbling group of permutations \sigma\colon \mathbb Z^d \to \mathbb Z^d with bounded range). This is equivalent to asking whether the action (\mathbb Z/2\mathbb Z)^{(\mathbb Z^d)} \rtimes W(\mathbb Z^d) \curvearrowright (\mathbb Z/2\mathbb Z)^{(\mathbb Z^d)} is amenable. The d = 1 and d = 2 and have been settled respectively in [9, 11]. By [12], a positive answer to this question would imply the amenability of the IET group. In this work, we give a partial answer to this question by introducing a natural strengthening of the notion of extensive-amenability which we call diffuse-extensive-amenability . Our main result is that for any bounded degree graph X , the action W(X)\curvearrowright X is diffuse-extensively amenable if and only if X is recurrent. Our proof is based on the construction of suitable stochastic processes (\tau_t)_{t\geq 0} on W(X)\, <\, \mathfrak{S}(X) whose inverted orbits \bar O_t(x_0) = \{x\in X\colon \text{there exists } s\leq t \text{\ s.t.\ } \tau_s(x)=x_0\} = \bigcup_{0\leq s \leq t} \tau_s^{-1}(\{x_0\}) are exponentially unlikely to be sub-linear when X is transient. This result leads us to conjecture that the action W(\mathbb Z^d)\curvearrowright \mathbb Z^d is not extensively amenable when d\geq 3 and that a different route towards the (non-?)amenability of the IET group may be needed.

Action-induced nested classes of groups and jump (co)homology

Abstract Using fixed-point-free group actions, we set up a scheme to define nested classes of groups indexed over ordinals. Restricting to cellular actions on CW-complexes, we find new classes of groups as well as new characterizations for some well-known classes. We extend some of the properties of the cohomological dimension of a group to groups with jump (co)homology and study their implications to cellular actions on finite dimensional CW-complexes that satisfy a quite general homological condition.

Relative property A and relative amenability for countable groups

We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if G has property A relative to a family of subgroups H and if each H∈H has property A, then G has property A. This result leads to new classes of groups that have property A. In particular, groups are of property A if they act cocompactly on locally finite property A spaces of bounded geometry with any stabilizer of property A. Specializing the definition of relative property A, an analogue definition of relative amenability for discrete groups is introduced and similar results are obtained.

Some new results on 1‐rotational 2‐factorizations of the complete graph

AbstractIt is known that a necessary condition for the existence of a 1‐rotational 2‐factorization of the complete graph K2n+1 under the action of a group G of order 2n is that the involutions of G are pairwise conjugate. Is this condition also sufficient? The complete answer is still unknown. Adapting the composition technique shown in Buratti and Rinaldi, J Combin Des, 16 (2008), 87–100, we give a positive answer for new classes of groups; for example, the groups G whose involutions lie in the same conjugacy class and having a normal subgroup whose order is the greatest odd divisor of |G|. In particular, every group of order 4t+2 gives a positive answer. Finally, we show that such a composition technique provides 2‐factorizations with a rich group of automorphisms. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 237–247, 2010

Geometric equivalence of groups

Based on the notion of geometric equivalence of groups, new classes of groups, namely, geometric varieties of groups, are defined. Some properties of such classes, including their relation to quasi-varieties and prevarieties of groups, are studied. Examples of torsion free nilpotent groups that are geometrically nonequivalent to their minimal completions, as well as an example of centrally metabelian groups that are geometrically nonequivalent but generate equal quasi-varieties, are given.

Nilpotent 1-factorizations of the complete graph

For which groups G of even order 2n does a 1-factorization of the complete graph K2n exist with the property of admitting G as a sharply vertex-transitive automorphism group? The complete answer is still unknown. Using the definition of a starter in G introduced in 4, we give a positive answer for new classes of groups; for example, the nilpotent groups with either an abelian Sylow 2-subgroup or a non-abelian Sylow 2-subgroup which possesses a cyclic subgroup of index 2. Further considerations are given in case the automorphism group G fixes a 1-factor. © 2005 Wiley Periodicals, Inc. J Combin Designs

Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity

A natural interpretation of automorphisms of one-rooted trees as output automata permits the application of notions of growth and circuit structure in their study. New classes of groups are introduced corresponding to diverse growth functions and circuit structure. In the context of automorphisms of the binary tree, we discuss the structure of maximal 2-subgroups and the question of existence of free subgroups. Moreover, we construct Burnside 2-groups generated by automorphisms of the binary tree which are finite state, bounded, and acyclic.

Cotilting and a Hierarchy of Almost Cotorsion Groups

We describe the structure of all cotilting and partial cotilting groups, and of all cotilting torsion-free classes of groups. Under V=L, we also describe all tilting and partial tilting groups. As a by-product, we introduce new classes of groups—called almost cotorsion groups. Unlike the well-known cotorsion groups, the almost cotorsion ones have a rather complex structure: we prove that any cotorsion free ring can be realized as an endomorphism ring of an almost cotorsion group.

New classes of groups containing Menon difference sets

New classes of groups containing Menon difference sets

New Classes of Groups Containing Menon Difference Sets

A theorem due to Davis on the existence of Menon difference sets in 2-groups is generalised to non-2-groups. The existence of Menon difference sets in many new non-abelian groups is established.

Torsion free groups

In this paper we introduce the class of torsion free k k -groups and the notion of a knice subgroup. Torsion free k k -groups form a class of groups more extensive than the separable groups of Baer, but they enjoy many of the same closure properties. We establish a role for knice subgroups of torsion free groups analogous to that played by nice subgroups in the study of torsion groups. For example, among the torsion free groups, the balanced projectives are characterized by the fact that they satisfy the third axiom of countability with respect to knice subgroups. Separable groups are characterized as those torsion free k k -groups with the property that all finite rank, pure knice subgroups are direct summands. The introduction of these new classes of groups and subgroups is based on a preliminary study of the interplay between primitive elements and ∗ \ast -valuated coproducts. As a by-product of our investigation, new proofs are obtained for many classical results on separable groups. Our techniques lead naturally to the discovery that a balanced subgroup of a completely decomposable group is itself completely decomposable provided the corresponding quotient is a separable group of cardinality not exceeding ℵ 1 {\aleph _1} ; that is, separable groups of cardinality ℵ 1 {\aleph _1} have balanced projective dimension ≤ 1 \leq 1 .

The balanced-projective dimension of abelian 𝑝-groups

The balanced-projective dimension of every abelian p p -group is determined by means of a structural property that generalizes the third axiom of countability. As a corollary to our general structure theorem, we show for λ = ω n \lambda = {\omega _n} that every p λ {p^\lambda } -high subgroup of a p p -group G G has balanced-projective dimension exactly n n whenever G G has cardinality ℵ n {\aleph _n} but p λ G ≠ 0 {p^\lambda }G \ne 0 . Our characterization of balanced-projective dimension also leads to new classes of groups where different infinite dimensions are distinguished.

The dimension subgroup problem

Whether the terms of the lower central series of a group are associated with the powers of the augmentation ideal of its integral group ring is a question which has been open for over thirty years. This question, the dimension subgroup problem, has achieved notoriety for the number of false proofs it has elicited. In this paper, we record general observations on the problem, verify the conjecture for several new classes of groups and develop generalizations which indicate new lines of research. In a subsequent paper [20], we examine the subgroups attached to group rings over arbitrary coefficient domains. The general properties of dimension subgroups are set forth in the first section. Several equivalent formulations of the conjecture are given. The subgroups are calculated in a number of special cases. A minimal counterexample is a finite p-group with cyclic center. In the second section, we take the approach that such reductions make available. The conjecture is established for Abelian-by-cyclic groups and for split extensions of Abelian groups by groups satisfying the conjecture; it follows inductively that the Sylow subgroups of the symmetric groups satisfy the problem and so every p-group embeds in a p-group for which the conjecture is verified. This analysis provides an unexpected bridge to the work of Passi who conjectured a strong result about the properties of factor sets of extensions of divisible groups byp-groups and showed how it would establish the dimension subgroup conjecture. Here another way of using Passi’s conjecture to solve the problem is introduced and applied in a minimal counterexample to obtain new results. The third section places the problem in a wider context: I f G, is associated with I”, then G, is associated with even larger ideals. It may be that these ideals will have to be identified before the dimension subgroup problem can be resolved affirmatively. Some such ideals are discovered by expanding the methods of Passi and Moran. The results obtained are used to establish the