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:

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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.

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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.

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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.

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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.

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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

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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

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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.

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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.

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.

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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.

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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

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].

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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.

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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.

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“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).

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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 .

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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.

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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].