A factorization is called normalized if each of its factors is a normalized subset. A subset A of G is defined to periodic if there is a g ∈ G \ {e} for which gA = A. A factorization is defined to be periodic if at least one its factors is periodic. A **finite** **abelian** group possesses **the** Haj´os **k**-**property** if it admits only periodic factorizations into **k** factors. For a subset A of G **the** notation hAi stands for **the** smallest subgroup of G that contains A, that is, hAi denotes **the** span of A in G. A normalized subset A of G is called a full-rank subset if hAi = G. In other words **the** normalized subset A of G is a full-rank

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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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Finally, let G be a group that satisfies **the** hypotheses of (e) and suppose first that G/Z(N ) is nilpotent. We claim that X := Z(N ) ≤ Z(G), and assuming this false we have by residual nilpotency that [X, G, G] < [X, G], so that we may factor and hence assume that [X, G, G] = 1. Choose an element a of X that has order p modulo Z(G), and consider **the** homomorphism θ : G → [X, G] given by θ(g) = [a, g] for all g ∈ G. **The** image of θ has exponent exactly p, but G/ ker(θ) is divisible and we have a contradiction that establishes **the** claim. It remains only to show that G/X is residually nilpotent, for then by induction on **the** class of N we may assume G/X nilpotent and hence G nilpotent, by **the** above. Since G/N is a divisible Baer p-group and hence **abelian**, there is a set {**K** λ : λ ∈ Λ} of G-invariant subgroups of N **with** trivial intersection such that each G/**K** λ is

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O Estado ajustador, em um confronto explícito como o padrão do Estado democrático, isenta-se, pro- gressivamente, do seu papel de garantidor de direitos, concretizando-se, assim, um encolhimento de suas responsabilidades sociais. Com efeito, um fenômeno- chave nessa nova ordem do capital é a desconstrução dos direitos que assume, então, uma dupla dimensão: a primeira é a erosão real dos direitos. Nesse contexto, direitos do trabalho são destituídos e **redei** nidos; di- reitos sociais não conseguiram consolidar suas garan- tias; direitos historicamente conquistados são negados ou fragmentados, coni gurando-se, desse modo, o desmonte dos serviços sociais. A outra dimensão é a erosão da própria noção de direitos e das referências pelas quais os direitos podem ser formulados. É o en- colhimento do horizonte da legitimidade dos direitos. Vivencia-se a dii culdade ou a impossibilidade de se- rem formulados como demanda, experimentando-se, mesmo, um mal-estar na enunciação dos direitos. E, mais, na ótica da valorização do capital, muitas vezes são tratados como ônus, “contabilizados” – tal como aparece na expressão “custo Brasil”, por exemplo – ou “l exibilizados”, na gramática neoliberal dominante. No mundo do trabalho, vem se operando um esva- ziamento da dimensão fundante do direito; ou seja, o direito deixa de ser uma prerrogativa de todos e se transforma em elemento indexado ao desempenho individual, como “recompensa”. É o direito metamor- foseado em benefício, objeto de retórica, capturado pela racionalidade do Mercado, que retira a “alma do direito”. E, nesta perspectiva, ocorre uma transferência de responsabilidades sociais, também para a socieda- de civil, fazendo ganhar espaço fenômenos como a

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[14] P. B. Kleidman and M. Liebeck,**The** Subgroup Structure of **the** **Finite** Classical **Groups**, London Mathematical Society Lecture Note Series 129, Cambridge University Press, Cambridge, 1990. MR1057341(91g:20001). Zbl 0697.20004. [15] V. Land´ azuri and G. M. Seitz, On **the** minimal degrees of projective

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4-manifold, [M]. For any :$ cohomologous to :, any ;$ cohomologous to ;, and any g : M Ä W(:$) homotopic to f, we have g*( ;$)([M])= f *( ;)([M]). Quinn then takes a **certain** weighted sum of f *( ;)([M]) over an arbitrary set of representatives of all homotopy classes of maps f. For a precise definition of **the** weights see [42]. As a matter of fact, Quinn only works out concretely his very abstract construction, which he defines in any dimension, for **the** classifying space of a **finite** group and cocycles on that space. In dimension four that corresponds to **the** restricted case in which H=[1]. **The** interesting invariants in dimension 4, that we have sketched above, were never considered by Quinn, or anyone else, explicitly. It is clear, by **the** arguments following our conjecture, that our state-sum invariants for a given semi-weak monoidal structure are equal to Quinn's invariants for **the** corresponding Postnikov invariants. This shows immediately that our state-sum invariants are homotopy invariants, rather than PL invariants. This is not surprising given **the** fact that we use **finite** **groups**. In dimension 3, **the** DijkgraafWitten invariants [29] are homotopy invariants as well. As already mentioned, **the** TuraevViro invariants are real homeomorphism invariants, but they require **the** use of quantum **groups** instead of **finite** **groups**. **The** categorical construction that underlies **the** DijkgraafWitten and **the** TuraevViro invariants is **the** same though; it is **the** specific input in that construction that makes **the** dif- ference. It is therefore reasonable to look for categorifications of **the** quan- tum **groups**, **the** representations of which could be **the** right input in our construction, presented in [37], for obtaining true PL invariants. This idea led Crane and Frenkel [23] to **the** definition of a Hopf category in **the** first place.

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Machine and tools elements made of **the** steel C90 and HS 6-5-2 immediately after **the** conventional hardening, need **the** tempering process. During **the** tempering, there is a transforma- tion of retained austenite into martensite and carbides release in martensite, what leads to **the** increase hardness of **the** high-speed steel, whereas in non-alloy steels tempering cause decrease **the** hardness.

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Recently, it has been shown [1,2] that large back-to-back correlations (BBC) of particle-antiparticle pairs of bosonic particles might appear in high energy nucleus-nucleus colli- sions as a consequence of in-medium mass modification of **the** bosons. Detailed calculations indicate that **the** BBC signal appears for values of transverse momenta below 1–2 GeV/c. More recently, it was shown [3] that BBC of similar strength might appear for fermionic particles as well. **The** main physical ingredient used in **the** evaluation of **the** effects of in-medium modified masses on two particle correlation functions is a quantum-mechanical correlation induced by a nonzero overlap between in-medium and free states. **The** induced quantum mechanical correlation can be represented in terms of two- mode squeezed states of **the** asymptotic, observable states and is implemented through a Bogoliubov-Valatin transformation. **The** possibility of measuring a significant BBC signal in heavy-ion collisions opens new interesting possibilities for accessing **the** properties of **the** matter formed in such collisions. **The** BBC signal is linked to in-medium mass modifications of hadrons in **the** hot and dense environment **the** detected particles experience before freezing out and in this sense BBC measurements provide independent pieces of information on medium modifications from **the** ones obtained from dilepton yields and spectra. However, there are several additional physical effects that interfere **with** mass modifications of **the** detected particles in **the** interpretation of **the** BBC signal.

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In Crypto 2000 [9],Ko et al proposed a new public key cryptosystem based on Braid **groups** which are non- **abelian**; they have used **the** search problems to build **the** Cryptosystem.Then Paeng et al [13] proposed a New Public Key Cryptosystem using **the** Discrete Log Problems in inner automorphisms of **the** semi-direct product of special linear **groups** (non-**abelian**) **with** Zp. **The** above system was named as MOR cryptosystem.Using Unitriangular[3], and Unimodular matrices, Ayan Mahalonobis[4] modified theMOR cryptosystem .

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symmetry with N = 2 of the Higgs Abelian model and. the appearane of Bogomol'nyi equations[r]

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c can be taken as 0.025 for n > 792000. This settled **the** question as to whether γ(G) grows linearly in n, rather than growing more slowly as a constant times ϕ(n). Proof of **the** improved lower bound in [7] relies on **certain** number theoretic results (see [5]), and **the** value of c obtained is unrealistically small because of **the** many approximations needed first to obtain and next to apply those results. Thus, despite **the** innovative methods used to achieve **the** definitive result that γ(S n ) and γ(A n ) grow

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This research was partially supported by FCT (Portugal) under **the** project PTDC/MAT-GEO/0675/2012, **the** fellowship SFRH/BPD/111131/2015 and by CMUP (UID/MAT/00144/2013), which is funded by FCT **with** national (MEC) and European structural funds through **the** programs FEDER, under **the** part- nership agreement PT2020.

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Later a major motivation arose from **the** natural sciences, in particular physics, where **certain** configurations are expressed geometrically in terms of numerical functions and coordinates, which may depend on chosen viewpoints and coordinate systems. **The** change of coordinate systems can then be described by a transformation group, acting on those functions, and **the** “true, objective” physical entities turn out to be symmetry classes or “orbits” of those functions under **the** action of **the** transformation group.

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Heritability estimates obtained **with** **the** two RRM (4,6,vf6 and 6,6,vf6) and by univariate analysis are pre- sented in Figure 3. **The** heritability estimates obtained **with** **the** TDMO ranged from 0.09 (12 th week) to 0.32 (42 nd week), **with** higher estimates at **the** end of lactation, proba- bly due to **the** lower number of records at these points. Heritabilities estimated by **the** 6,6,vf6 model ranged from 0.09 to 0.26, while those estimated by **the** 4,6,vf6 model ranged from 0.09 to 0.21. For both RRM, **the** estimates were lower during mid-lactation, from 14 th to 30 th week. In general, RRM heritability estimates were similar to those obtained by TDMO, **with** larger differences in test day milk yields around 14 th to 28 th weeks. **The** estimates obtained by 6,6,vf6 model oscillated more than those from 4,6,vf6 model, **with** higher heritabilities observed at **the** extremes of lactation. Trends of **the** heritability estimates, obtained by RRM, followed those reported in **the** literature, i.e., higher estimates at **the** beginning and towards **the** end of lactation (Strabel and Misztal, 1999; Tijani et al., 1999; Brotherstone et al., 2000). Difficulties to model **the** vari- ances at **the** extremes of lactation can be explained, in part, by **the** biological processes that occur at **the** beginning of lactation and **the** smaller number of records at **the** end. For Zebu and tropical native breeds, both reasons can be even more important. In general, these animals have not been as intensively selected for milk production as Holsteins and population sizes are smaller. It is usually necessary to keep **the** calf **with** **the** dam during milking and short lactations are frequent.

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by some authors. Recently, Podoski and Szegedy proved **the** truth of **the** converse of Schur’s theorem for capable **groups**. They also established an explicit bound for **the** index of **the** center of such **groups**. This paper is devoted to determine some families of **groups** among non-capable **groups** which satisfy **the** converse of Schur’s theorem and at **the** same time admit **the** Podoski and Szegedy’s bound as **the** upper bound for **the** index of their centers.

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