Ker = (g G: (g) = (1) Characters are class functions i.e. they take a constant value on a given conjugacy class. Isomorphic representations have the same characters and if a representation is the di- rect sum of subrepresentations, then the cor- responding **character** is the sum of the char- ABSTRACT

Proof. As G is nonsolvable, Proposition 2 implies that B(G) is disconnected. Since B(G) is a union of paths, we deduce that ∆(G) is triangle-free. Now [21, Theorem A] implies that | ρ(G) | ≤ 5. Also [21, Lemma 4.1] verifies that there exists a **normal** **subgroup** M of G such that G/N is an almost simple group with socle M/N . Furthermore ρ(M) = ρ(G). Now it follows from Ito-Michler and Burnside’s p a q b theorems that | ρ(M/N ) | ≥ 3, so | ρ(G) | ≥ | ρ(M/N ) | ≥ 3. Hence 3 ≤ | ρ(G) | ≤ 5. In Lemma 5 we observe that if | ρ(G) | = 5, then G ≃ P SL(2, 2 n ) × A, where A is an abelian group and | π(2 n ± 1) | = 2. This implies that n(B(G)) = 3. On the other hand, since G is a nonsolvable group, by [10, Theorem 6.4] we conclude that n(B(G)) = 3 if and only if G ≃ P SL(2, 2 n ) × A, where A is an abelian group and n ≥ 2. So we may assume that | ρ(G) | ≤ 4 and n(B(G)) = 2. As n(Γ(G)) = n(B(G)) = 2, [10, Theorem 7.1] implies that one of the connected components of Γ(G) is an isolated vertex and the other one has diameter at most two. Therefore | cd(G) | ∈ { 4, 5 } .

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| G/K | | | Out(K/H ) | = 2 or 3. Thus | H | | 2 2 · 3 · 7 2 · 19 or | H | | 2 2 · 7 2 · 19 and hence, Lemma 1.7 guarantees that H is solvable. Therefore, an easy calculation and Lemma 1.4 show that a 19-Sylow **subgroup** of H is **normal** in H and so, it is **normal** in G. Thus G has a **normal** abelian 19-Sylow **subgroup** and hence, Ito’s theorem implies that χ(1) = 2 3 · 3 · 19 | [G : P ], which is impossible. Therefore K/H ∼ = L 3 (7) and hence, G = K ∼ = L 3 (7), as desired.

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would have an abelian **normal** Sylow p-**subgroup** against our assumption. So P Y /Y is a nontrivial abelian **normal** **subgroup** of K/Y ∼ = L and this is possible only if N ∼ = P Y /Y is an elementary abelian p-group. In this case N = soc(L) has a complement, say T , in L. In particular { (t, . . . , t) ∈ L t | t ∈ T } is a complement of N t in L t and all the minimal **normal** subgroups of L t are T -isomorphic to N.

This interpretation clari ﬁ es also the enigmatic fact that, in randomised clinical trials, patients allocated to placebo consistently exhibit a higher survival rate when they are adherent, 46,47 leading to explanations using the “ healthy adherer ” concept for this fascinating observation. 46 The data reported herein support this explanation by demon- strating a strong link between adherence to medication and dietary and exercise recommendations and other protective behaviours. We suggest that **character** traits such as patience and obedience and, possibly, others such as ﬁ de- lity to habits, optimism, joy and caution 29 have a real causal effect, leading, when they are present, to a “ healthy adherer ” phenotype. By “ real causal effect ” , we mean that it is not only a statistical link between observations but a mechanism, in the same sense that insulin causes a decrease in blood glucose concentration. 26,30 The absence of these positive **character** traits leads to conditions of what we propose to dub a “ nonadherence syndrome ” :

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Since the level subsets of η and µ are subgroups of G, η and µ are L-subgroups of G. As η ⊆ µ, η is an L-**subgroup** of µ. Now in view of the fact that every **subgroup** of a nilpotent group is nilpotent, it follows that all the level subsets of η are nilpotent subgroups of the corresponding level subsets of µ. Therefore the converse of Theorem 4.1[3], implies that η is a nilpotent L-**subgroup** of µ.

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contradiction shows that F/H is finite. Put B/H = (P/H) p . If P/B is infinite then P/B is not finitely generated. Therefore the cocentralizer of P in A is a noetherian R-module. By corollary 5 G = N D(G). Since G/P is finite, it follows that the cocentralizer of G in A is a noetherian R-module by lemma 1. This contradiction proves that (P/H)/(B/H) is finite. By lemma 3 [9] P/H = (V /H) × (D/H) where D/H is divisible and V /H is finite. D is a G-invariant **subgroup**. Put K = D. Since G/D is finite, it is suffices to apply lemma 4.

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Given an MO-group, G , we s h a l l suppose that G s p l i t s over A where ^ i s a d i v i s i b l e , **normal**, abelian **subgroup** of G and GIA i s a b e l i a n . So there i s a homomorphism from GlA into aut(4) which defines the extension of A by GlA (see Kurosh [ 1 7 ] , p. 149). The f i r s t change we make i s to denote by $ the homomorphic image of G/A in aut(>l) Hence, f o r the remainder of t h i s t h e s i s , we s h a l l be discussing semi-direct products of abelian 0-groups by abelian 0-automorphism groups. 0 . 3 Theorem 6 w i l l provide the link between these and the MO-groups under consideration.

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Abstract: In this paper, we have introduced the concept of step N-fuzzy soft **subgroup** and step N-fuzzy soft cosset of a given group. We study the concept of step N-fuzzy soft **normal** **subgroup** and have discussed various related properties. We have also studied the effect on the image and inverse image of step N-fuzzy soft **subgroup** (**normal**) under group homomorphism.

In the beginning we prove that SL(4,R) is the derived group of GL(4,R). We use the result that if x, y are nn matrices then det (xy) = det(x)det(y). This **relation** asserts that mapping x det(x) (xGL(4,R)) is a homomorphism of GL(4,R) into R*, the multiplicative group of non-zero real numbers.

Each flour was examined at the absorption chosen in the bakehouse which we have called “**normal**,” and at four other water contents ±§ gal- lon and ±1 gallon per 280-lb. sack of flour.11 Each dough was fermented for four hours, at the end of which time samples were taken for viscosity and modulus measurements. In figure 2 curves nre drawn showing the relationship between these two properties at each of the five water con-

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[iii] Why only the nobles can affirm themselves, and therefore create values Yet this last point, of course, begs the question as to why Nietzsche thinks this is the case. That is, it begs the question as to why he believes only noble, ‘higher’ types, are capable of affirming their mode of existence through the creation of values. And it is hoped then that we can reveal more about the link between ‘affirmation’ and value creation, and therefore slave perversion, by looking at this issue. For, doubtless to begin with, the answer has something to do with the specific capacity of the nobles for self-affirmation. And this, we can say, has typically been interpreted in a ‘passive’ sense. This is because it has been argued that only the nobles are capable of value-creating self-affirmation as it is only they, rather than the slaves, who can stand in a genuinely affirmative **relation** to life. It is then the privileged conditions of their lives which, on this view, uniquely allow them to affirm themselves. As Conway says, ‘The designation good...originated with the nobles’ selfish or egoistic assertion of their own incomparable self-worth and unrivalled social station.’ 80 Put another way, it is the noble’s sense of self-worth, based on privilege and social superiority, which allows them to affirm themselves and their lives as ‘good’. Conversely, we can say, the opposite is true for the downtrodden. That is, for those whose lives are characterised by misery and subordination it is their unhappiness which equally prevents them stamping on their mode of existence any positive value.

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Todd - Coxeter algorithm. In this implementation cosets may be enumerated by the Lookahead or Felch method. The TC program also provides for the manipulation of partial and complete coset tables. This version also contains an implementation of the Reidemeister - Schreier algorithm, added by E.F.Robertson, which enables the user to obtain defining relations for a **subgroup** H of finite index. In Chapters 4 , 5 and 6 we have used the TC

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to expect meaningful results on groups beyond the class of generalized soluble groups. We will study our groups under the following restriction. Recall that a group G is said to be radical if it has an ascending series whose factors are locally nilpotent. Some standard properties of radical groups can be found in [9]. In this paper we start the consideration of radical groups, whose non-finitely generated subgroups are transitively **normal**. Recall that a periodic radical group is locally soluble. But converse is not true.

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Abstract: Chronic kidney disease (CKD) and **its** associated morbidity pose a worldwide health problem. As well as risk of endstage renal disease requiring renal replacement therapy, cardio- vascular disease is the leading cause of premature death among the CKD population. Proteinuria is a marker of renal injury that can often be detected earlier than any tangible decline in glom- erular filtration rate. As well as being a risk marker for decline in renal function, proteinuria is now widely accepted as an independent risk factor for cardiovascular morbidity and mortality. This review will address the prognostic implications of proteinuria in the general population as well as other specific disease states including diabetes, hypertension and heart failure. A variety of pathophysiological mechanisms that may underlie the relationship between renal and cardiovascular disease have been proposed, including insulin resistance, inflammation, and endothelial dysfunction. As proteinuria has evolved into a therapeutic target for cardiovascular risk reduction in the clinical setting we will also review therapeutic strategies that should be considered for patients with persistent proteinuria.

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heat entering the workpiece to the total heat). Residual tensile stresses, which are primarily thermal in origin, may be unacceptable. Investigations have found that preferred compressive stresses are more likely to be achieved with CBN and diamond grinding wheels. Results of investigations [3] indicate an advantage of CBN and diamond grinding is a smaller proportion of the energy entering the workpiece. The partition ratio is therefore a useful indicator of grinding-wheel performance relevant to the likelihood of tensile stresses. Most of the energy enters the workpiece (90%) when grinding conventional roll-bearing steels using an alumina grinding wheel ([3] and [4]). This is given by kinematics conditions and the fact that the thermal conductivity of conventional roll-bearing steels (46 W/mK) is higher than that the alumina grinding wheel (6¸30 W/mK, wide range of the presented values). The heat distribution when grinding a VT 9 titanium alloy differs from the heat distribution when grinding conventional roll-bearing steels because of the poor thermal properties of titanium alloys (the thermal conductivity of titanium alloys is 7.5 W/mK) and so this paper deals with **its** analysis and the **relation** to quality of the ground parts in terms of residual stresses.

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The survey identified that the practicum is not given due value, so the ministry of education and education faculties should take some measures to communicate the significance of practicum because without realizing **its** importance, TE goals may be at stake. The least mean scores were for teacher educators, but because the teacher educators themselves were not included in the survey, so it might have affected the scores. After teacher educators, the least mean scores were for STs, who were the main part of the survey and the main stake holders. The teacher education faculties should give special attention to this issue. They should communicate the real importance of practicum for teacher development and the process of practicum should be given more weightage in the whole programme. Still, currently it has only less than 4% weightage (Chen & Mu 2010) although 33% has been recommended in the literature (Quick & Sieborger, 2005).

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Chapters 3–6 investigate other higher-order boundary value problems having only space singularities which appear most frequently in literature. They provide existence results for (n, p)-problems, conjugate problems, Sturm-Liouville problems, and Lidstone problems. Part II consists of Chapters 7–11 and deals with scalar second-order singular bound- ary value problems with one-dimensional φ-Laplacian. The exposition is focused mainly on Dirichlet and periodic problems which are considered in Chapters 7 and 8, respec- tively. Section 7.1 is fundamental for further investigation. The operator representation of the regular Dirichlet problem with φ-Laplacian is derived here and the methods of a priori estimates and lower and upper functions are developed. In Sections 7.2–7.4, three existence principles are presented. These principles together with the principles of Chapter 1 are then specialized to important particular cases and existence theorems and criteria extending and supplementing earlier results are obtained. Section 7.2 deals with time singularities, Section 7.3 with space singularities, and Section 7.4 with mixed singularities, that is, both time and space ones. In Chapter 8, we consider the existence of periodic solutions. We start with the method of lower and upper functions and with **its** relationship to the Leray-Schauder degree in Section 8.1. Section 8.2 is devoted to prob- lems with a nonlinearity having an attractive singularity in **its** first space variable. Sec- tions 8.3 and 8.4 deal with problems with strong and weak repulsive space singularities, respectively. An existence theorem for periodic problems with time singularities is given in the last section of Chapter 8. In Chapter 9, we study two singular mixed boundary value problems. The latter arises in the theory of shallow membrane caps and we discuss **its** **solvability** in dependence on parameters which appear in the di ﬀ erential equation. In Chapter 10, we treat problems which may have singularities in space variables. Boundary conditions under discussion are generally nonlinear and nonlocal. We present general principles for **solvability** of regular and singular nonlocal problems and show some of their applications. Chapter 11 is devoted to a class of problems having singularities in space variables. Implementation of a parameter into the equation enables us to prove **solvability** of problems with three independent (generally nonlocal) boundary condi- tions. We deliver an existence principle and **its** specialization to the problem with given maximal values for positive solutions.

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assume that M is nonabelian. It follows that if M/N is a chief factor of G 0 , then M/N ∼ = S k , where S is a nonabelian simple group and k ≥ 1. By Lemmas 2.4 and 2.5, M/N has a nontrivial irreducible **character** ϕ which extends to G 0 and so if ψ ∈ Irr(G 0 /M ) with ψ(1) = 2 24 , then by Gallagher’s Theorem, we deduce that G 0 has an irreducible **character** of degree ψ(1)ϕ(1) = 2 24 ϕ(1) > 2 24 . However this is impossible as 2 24 ϕ(1) divides no **degrees** of G. Hence M = 1.