Top PDF On finite groups with square-free conjugacy class sizes

On finite groups with square-free conjugacy class sizes

On finite groups with square-free conjugacy class sizes

In parallel to the research on the effect of square-free class sizes on a group structure, the study of groups factorised as the product of subgroups was also gaining an increasing interest, especially when they are connected by certain permutability properties. A new research line arises when considering both perspectives simultaneously, that is, to analyse factorised groups in which the conjugacy class sizes of some elements in the factors are square-free. This new tendency has been little investigated, and perhaps the first authors that tested some preliminary results were Liu, Wang and Wei:
Show more

8 Read more

Conjugacy classes and graphs of two-groups of nilpotency class two

Conjugacy classes and graphs of two-groups of nilpotency class two

Besides, the studies on the influence of the size of the conjugacy classes on the structure of a finite group have been the subject of research over the years. Many researchers produced papers on this topic, for instance [10–17]. However, very little is known about how the conjugacy class sizes depend on the order of the commutator subgroup.

24 Read more

Orbit counting in conjugacy classes for free groups acting on trees

Orbit counting in conjugacy classes for free groups acting on trees

Abstract. In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Further, we assume an irrationality condition on the edge lengths. We obtain an asymptotic for the number of elements in a fixed conjugacy class for which the associated displacement of a given base vertex in the universal covering tree is at most T . Under a mild extra assumption we also obtain a polynomial error term.

14 Read more

Conjugacy classes contained in normal subgroups: an overview

Conjugacy classes contained in normal subgroups: an overview

Abstract. We survey known results concerning how the conjugacy classes contained in a normal subgroup and their sizes exert an influence on the normal structure of a finite group. The approach is mainly presented in the framework of graphs associated to the conjugacy classes, which have been introduced and developed in the past few years. We will see how the properties of these graphs, along with some extensions of the classic Landau’s Theorem on conjugacy classes for normal subgroups, have been used in order to classify groups and normal subgroups satisfying certain conjugacy class numerical conditions.
Show more

14 Read more

Groups with reality and conjugacy conditions

Groups with reality and conjugacy conditions

Finally in Section 5 we study F C -groups with two non-trivial finite class sizes. Ito ([14]) proved that finite groups with this property are soluble. More precise results on the structure of such groups have been obtained by Camina ([3]) and Dolfi and Jabara ([8]). We prove the following result: Theorem 1.4. Let G be a periodic F C-group with two non-trivial conjugacy class sizes. Then G = N M, where N is normal in G and N, M are nilpotent. In particular, G is soluble.

14 Read more

Bipartite divisor graph for the set of irreducible character degrees

Bipartite divisor graph for the set of irreducible character degrees

The notion of the bipartite divisor graph was first introduced by Iranmanesh and Praeger in [4] for a finite set of positive integers. As an application of this graph in group theory, in [2], the writers considered this graph for the set of conjugacy class sizes of a finite group and studied various properties of it. In particular they proved that the diameter of this graph is at most six, and classified those groups for which the bipartite divisor graphs of conjugacy class sizes have diameter exactly 6. Moreover, they showed that if the graph is acyclic, then the diameter is at most five and they classified the groups for which the graph is a path of length five. Similarly, Taeri in [20] considered the case that the bipartite divisor graph of the set of conjugacy class sizes is a cycle and (by using the structure of F -groups and the classification of finite simple groups) proved that for a finite nonabelian group G, the bipartite divisor graph of the conjugacy class sizes is a cycle if and only if it is a cycle of length 6, and for an abelian group A and q ∈ { 4, 8 } , G ≃ A × SL 2 (q). Inspired by these papers, in this work we consider
Show more

11 Read more

Asymptotics in conjugacy classes for free groups

Asymptotics in conjugacy classes for free groups

Margulis [39] did not consider orbit counting in conjugacy classes; however, Parkkonen and Paulin considered orbit counting in conjugacy classes in the setting of variable negative curvature. In [42], Parkkonen and Paulin established asymptotic formulae for orbit counting in conjugacy classes in the setting of compact manifolds with pinched variable negative curvature. By pinched we mean the manifold’s sec- tional curvatures are bounded between two negative constants. They proved an exponential growth rate for the conjugacy counting function and gave an asymp- totic formula for finitely generated discrete groups of isometries of the hyperbolic plane. Indeed, in the special case that the group has co-finite volume Parkkonen and Paulin’s result is analogous to the result in Theorem 3.1.1 for metric graphs. Let Γ be a finitely generated non-elementary Fuchsian group and C a non-trivial conjugacy class in Γ. Parkkonen and Paulin showed that the associated conjugacy counting function N C (T) has an asymptotic formula N C (T) ∼ Ce hT /2 as T → ∞ and
Show more

105 Read more

Finite groups with the same conjugacy class sizes as a finite simple group

Finite groups with the same conjugacy class sizes as a finite simple group

Definition 2.9. [17] For a group H, the number of isomorphism classes of groups with the same set OC(H) of order components is denoted by h(OC(H)). If h(OC(H)) = k, then H is called k-recognizable by the set of its order components and if k = 1, then H is simply called OC- characterizable or OC-recognizable.

11 Read more

On residually finite semigroups of cellullar automata

On residually finite semigroups of cellullar automata

2.2. Residually finite semigroups. It is clear from the general definition of residual finiteness given in the Introduction that a group is residually finite as a group if and only if it is residually finite as a monoid and that a monoid is residually finite as a monoid if and only if it is residually finite as a semigroup.

7 Read more

Embeddings into Thompson's group V and coCF groups

Embeddings into Thompson's group V and coCF groups

Thus, as classes of languages become more complex, some “corresponding” classes of groups become wider, and thus harder to comprehend in a meaning- ful fashion. Ways to build these correspondences are through the word or co-word problems for groups, but other flavours of correspondence have also been seen (see, e.g., [2, 25, 19, 16, 17, 18]). To date, classifications of corresponding sets of groups (for classes of languages) only exist for very simple classes of languages, but the results on the group theory side are quite striking. In order to discuss this further, we need to give a definition.
Show more

15 Read more

Conjugacy and order classes of two-generator p-groups of nilpotency class two

Conjugacy and order classes of two-generator p-groups of nilpotency class two

GAP [11] is a system for computational discrete algebra, with emphasis on computational group theory. GAP provides a programming language, a library of functions that implement algebraic algorithms written in the GAP language as well as libraries of algebraic objects such as for all non-isomorphic groups up to order 2000.

16 Read more

Finite non-nilpotent groups with few non-normal non-cyclic subgroups

Finite non-nilpotent groups with few non-normal non-cyclic subgroups

the Sylow subgroups of G of odd order are non-normal in G. Thus all of their normalizers are cyclic. By Burnside’s theorem, G has a normal q-complement for each odd prime q ∈ π(G). The intersection of all of these normal q-complements is a Sylow 2-subgroup of G, a contradiction. □ Theorem 2.2. Let G be a finite non-nilpotent group with ν nc (G) = 1. If M is abelian, then G is

6 Read more

A note on maximal length elements in conjugacy classes of finite coxeter groups

A note on maximal length elements in conjugacy classes of finite coxeter groups

together with additional information, is tabulated in Section 3. These were calculated with the aid of Magma [1]. In the case of conjugacy classes of involutions, a rule to determine the minimal and maximal length in the class is given in [10], and a complete description of the set of elements of maximal and minimal length appears in [9].

10 Read more

Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes

Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes

Belyaev and Sesekin characterized minimal non-F C-groups when they have a non-trivial finite or abelian factor group. They proved that such minimal non-F C-groups are finite cyclic extensions of divisible abelian p-groups of finite rank, where p is a prime, hence they are Chernikov groups. In [4] it is proved, for groups having a proper subgroup of finite index, that the property of being a minimal non-(P F )C-group is equivalent to that of being a minimal non-F C -group. Note that in [4], the property (P F )C was denoted by P C. Since in [9], groups with polycyclic conjugacy classes have been considered and denoted by P C -groups, it is convenient, to denote in this note, groups with polycyclic-by-finite conjugacy classes by (P F )C-groups. In [9, Theorem A] it is proved that a non- perfect minimal non-P C -group is a minimal non-F C -group. Here we will generalize this last result to non-perfect minimal non-(P F )C-groups. Our first result is the following theorem.
Show more

7 Read more

Free subgroups of surface mapping class groups

Free subgroups of surface mapping class groups

The mapping class group MCG(Σ) of Σ is the group of all homotopy classes of orientation preserving self-homeomorphisms of Σ. It is a finitely presented group, generated by a finite collection of Dehn twists about simple closed curves of Σ. There is a natural action of MCG(Σ) on T (Σ) by changing the marking; the quotient T(Σ)/MCG(Σ) is the moduli space of Σ. Except for a few low-dimensional cases, the mapping class group MCG(Σ) is isomorphic to an index 2 subgroup of the full isometry group of both the Teichm¨ uller metric and the Weil-Petersson metric, by results of Royden [22] and Masur-Wolf [15] respectively.
Show more

13 Read more

Class sizes

Class sizes

“The evidence that we are looking at relates to classes—they should be reduced as a whole. Some local authorities have already achieved the proposals for class sizes that were set out by the previous Government by having more than one teacher for a larger class. It is up to local authorities to approach us if they think that that is an issue, and if they wish either to start to do that or to continue to do that….We should consider whether there are any merits in that—if that is how councils want to progress—but my preference is for class sizes as a whole to be reduced.” (Scottish Parliament Education, Lifelong Learning and Culture Committee 2007b).
Show more

14 Read more

Locally graded groups with a condition on infinite subsets

Locally graded groups with a condition on infinite subsets

Lemma 2.5. Let L be a finitely generated residually finite group which satisfies T ( ∞ ) and suppose that H / ∈ T for each subgroup H of finite index in L. Then there exists a normal subgroup G of finite index in L such that G = N ⟨ t ⟩ for some normal subgroup N of G and element t satisfying { ta, tb, tb } ∈ T for all a, b ∈ N .

7 Read more

On the Mark and Markaracter Tables of Finite Groups

On the Mark and Markaracter Tables of Finite Groups

Groups are often used to describe symmetries of objects. One goal of Group Theory is how the symmetry of a molecule is related to its physical properties and provides a method to determine the relevant physical information of the molecule. In other words, the symmetry of a molecule provides many important physical aspects and this is what makes group theory so powerful.

14 Read more

The exact number of conjugacy classes for 2 - generator p - groups of nilpotency class 2

The exact number of conjugacy classes for 2 - generator p - groups of nilpotency class 2

The Groups, Algorithms, and Programming (GAP) software is used in this research to gain insight into the 2-generator p-groups of class 2 (p an odd prime), to provide examples and to check the theoretical results obtained. GAP is a powerful tool and can be used to construct large p-groups and compute their conjugacy classes.

17 Read more

Character expansiveness in finite groups

Character expansiveness in finite groups

In this section we will list some basic results which will be needed in later parts of this paper. The paper [8] considered the following notion “weaker” than conjugacy expansiveness. We say that G is normal conjugacy expansive if for any normal subgroup N and any conjugacy class C of G the normal set N C consists of at least as many conjugacy classes of G as N does. Normal conjugacy expansive groups have been entirely described in [8].

9 Read more

Show all 10000 documents...