The SCRE literature review found that although a large volume of evidence existed on **class** **sizes**, and many of these studies were very good, “..questions remain about the research designs and the statistical analyses employed, and also the generalisability of the findings” (Scottish Executive 2006). The review also found that there was no relevant research on **class** **sizes** that had been carried out in Scotland. Much of the research was USA based, including the often quoted Student Teacher Achievement Ratio (STAR) project in Tennessee. Despite the large volume of research the findings have been mixed. The report described the findings over a number of years as “..at best confusing, sometimes even contradictory.”

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Abstract. A finite group G satisfies the one-prime power hypothesis for conjugacy **class** **sizes** if any two conjugacy **class** **sizes** m and n are either equal or have common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to S × A where A is abelian and S ∼ = P SL 2 (q) for q ∈ { 4, 8 } . We confirm this conjecture.

institutional level. As paragraphs 86 to 87 recognise, there are a number of different factors informing this debate and it would be wrong to reduce the multi-faceted nature of the decisions being made to a question of the impact on **class** **sizes**. However, the implementation of the new qualification structure does have implications for **class** **sizes**. Potentially, removing the expectation that students will move from four subject choices to three could alleviate the very prevalent issue of much smaller classes typically being offered in year 13 compared with year 12. However, the risk must be acknowledged that under a linear system more students may fail to complete their qualification if they do not have the opportunity to discontinue a subject in which they have not performed as well and without the reassurance of successfully gaining AS qualifications. Again, this may put pressure on the year 12 to 13 transition point and could place institutions under financial strain if more students drop out completely. 122. In making decisions about implementing the linear A levels, providers may wish to

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Proof. (i) By Lemma 2.5, | S | | | G/Z(G) | . Now let there exist q ∈ π(G/Z (G)) − π(S). Then since q ∤ | S | , we get that q does not divide any conjugacy **class** **sizes** of G and hence, Lemma 2.1 forces the q-Sylow subgroup of G to be a subgroup of Z(G), so q ̸∈ π(G/Z(G)), which is a contradiction. This yields that π(G/Z (G)) = π(S). If q ∈ π(G/Z (G)) = π(S) such that | G/Z(G) | q ̸ = | S | q , then

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parents’ choice of a school over another. Smaller **class** **sizes** are generally perceived as allowing teachers to spend more time with each student and less time in classroom management, thereby providing better instruction tailored to the students’ individual needs, and ensuring higher performance. In this respect, **class** size may be viewed as an indicator of the quality of a school system. At the lower secondary level and among all OECD countries with comparable data, the average **class** size varies from 20 students or fewer in Denmark, Estonia, Finland, Iceland, Luxembourg, Slovenia, Switzerland (public institutions) and the United Kingdom, to more than 34 students in Korea. The contrast is even more striking with other G20 countries which provided data (i.e. Argentina, Brazil, China, Indonesia and the Russian Federation); in China, for instance, the number of students per **class** reaches the 50 students mark (see Indicator D2 in OECD, 2012). It is worth noting that classes tend to be smaller in primary education, the number of students per **class** growing by two or more students between the primary and the lower secondary level. This trend has intensified between 2000 and 2010, particularly among countries that traditionally reported bigger **class** **sizes**, such as Japan and Korea. ( OECD, 2012)

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In contrast to what happens in Theorems 4.1 and 4.2 (and also in the corresponding ones in [18]), the hypotheses in Theorem 4.5 for the conjugacy **class** **sizes** of the elements x ∈ A ∪ B are not automatically acquired by the factors. For instance, the group G of Example 2.3 can be factorised as the mutually permutable product of A = D 10 × D 10 and B = [C 5 × C 5 ]C 3 (we checked this using GAP,

A **Class** **Sizes**, Staffing and Resources Working Group was established by the previous Scottish Executive in 2005. The interim and final reports included estimates of the training and salary costs for a number of scenarios including, reducing P1-3 to 25, reducing P1-7 to 20, providing additional CPD and employing more classroom assistants (Scottish Government, 2006, 2007b). Their final report, published after the 2007 election recommended a longitudinal study into the impact of **class** **sizes** should be commissioned and that the effects of the reduction in P1-P3 classes to a maximum of 18 should be evaluated as the reductions are rolled out. These recommendations have not been implemented.

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Last year the number of infant classes of over 30 pupils (and pupils within such classes) increased; there was a corresponding increase in the number of junior classes of over 30 pupils. The total number of infant and junior classes of over 30 pupils has risen a little over the past 7 years but remains lower than a decade ago when the statutory duties on local authorities in respect of infant **class** **sizes** were introduced. Chart 1: Percentage of Infant and Junior pupils in classes over 30

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In a recent paper [], the authors studied the structure of G under the condition that the largest two p-regular conjugacy **class** **sizes** (say, m and n) of G are coprime, where m > n and p n. Notice that, when G = N, the condition n dividing |N /Z(N)| is of course true, so our aim is, by eliminating the assumption p n, to investigate the properties of N under the corresponding condition. More precisely, we prove the following.

the sparseness of the data, minority data sets still can provide useful information if utilized appropriately. To demonstrate the usefulness of utilizing the minority data, we compare ADG with the OCC method developed in Park et al. (2010) using four sample data sets; this OCC method was proven to provide asymptotically the tightest bound for majority data points. For these four sample data sets, we select the training and test data such that the training data sets have the smallest value of imbalance ratio reported in Table 2. As Figure 4 shows, the OCC could be effective, for instance, duplicating ADG’s performance in the case of Pima data. One drawback is that OCC methods often suffer from a high false alarm rate, while attaining a high detection power (e.g. in the case of the Ionosphere data). When an OCC tries to build the tightest possible closed boundary around the majority data, the result can be an over-tightened boundary, instead of a boundary loose enough to identify all majority data points. On the other hand, in the two-**class** cases, the existence of minority data points can actually help relax the position of the decision boundary, at least locally where these minority data points are present. For more detailed comparisons of another OCC method with two-**class** classifiers, the reader may consult (Hempstalk et al., 2008); the results presented there also confirm the argument that if minority data are utilized, one generally observes an improvement in the minority detection.

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Abstract. We prove that every finite group G contains a three-generated subgroup H with the following property: a prime p divides the degree of an irreducible character of G if and only if it divides the degree of an irreducible character of H. There is no analogous result for the prime divisors of the **sizes** of the conjugacy classes.

Abstract: Tests should reflect teachers’ observations, classroom discussions and analysis of students’ work. The aim of this paper is to check whether these components are taken into account in the actual classroom tests organised in EFL classes in Benin. To this end, this paper uses a descriptive and quantitative approach to examine EFL in-service teachers’ views on the effects of tests on English teaching and learning. As a matter of fact, 100 in- service EFL teachers, i.e. 20 teachers per school, have been selected at random in five secondary schools in the Mono and Couffo regions, namely CEG1 Lokossa, CEG Comé, CEG1 Grand-Popo, CEG Djakotomey and CEG1 Azovè. On average, 80 percent of the questionnaires distributed have been filled and returned. The analyses carried out subsequently indicate that students poor performance in English tests is due to the lack of training for teachers as well as to the large **class** **sizes** and the lack of knowledge in test development and administration procedures. Furthermore, the lack of didactic materials in schools and students’ low motivation also contribute to their poor performance. The implications of this situation have been elucidated in a bid to enable various stakeholders to act to solve the problem.

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who reported an aversion to public speaking also reported a preference for smaller **class** **sizes** or smaller group activities. Unfortunately it is not possible to conclude whether or not this association is causal from the current data. It is feasible that smaller **class** **sizes** or increased small group activities may reduce students’ fear of speaking in public; however, further research is required to investigate this specific association. The current study did identify specific activities and teach- ing modalities that may assist students with large group participation. These include smaller groups and small group activities, involvement of other students, and use of audience response systems. Although it would be difficult to change the overall number of students within the lecture environment, there are ways of incorporating some facets of small group teaching into the large group setting; for example, team-based learning, breakout groups (pair share and buzz groups), and games or quizzes. 15,16 Audience response systems (“clickers”)

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someone which situation would result in a higher quality of instruction: a **class** with one instructor and 10 students, or, a **class** with one instructor and 50 students, it probably would not be a surprise if the answer is the former. Typically, the thought is that smaller **class** **sizes** allow for more meaningful student-to- instructor interaction and a higher quality of instruction. Despite its growth, there is still skepticism when it comes to online learning. Whether it is the quality of the course, lack of interaction (Maguire, 2005), the quality of the students enrolling (O'Quinn & Corry, 2002), or the quality of learning and/or instruction, online courses have been under scrutiny. In relation to **class** size, some instructors feel that the quality of instruction is questionable in online courses with larger **class** **sizes** (Parker, 2003; University of Illinois, 1999 as cited in Orellana, 2006).

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set X ∗ and two vertices m and n are connected if gcd(m, n) > 1. Motivated by [3], in this paper, we introduce the bipartite divisor graph for character degrees of a finite group and obtain some properties of this graph. The concept of the bipartite divisor graph for integer subsets has been considered in [4] and the bipartite divisor graph for group conjugacy **class** **sizes** has been studied in [3] and [5]. The bipartite divisor graph B (X) is the graph with vertex set the disjoint union 1 ρ(X) ∪ X ∗ and edge set

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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Those operational details are rarely the stuff of headlines but they matter greatly for headteachers and governing bodies as they attempt to balance the books. Our analysis finds that the department’s choices will cause schools with high levels of disadvantage or small **class** **sizes** to be less likely to receive enough additional funding to cover the cost of their teachers’ pay rise. The reason why disadvantaged schools and schools with smaller **class** **sizes** will receive less is complicated but illustrates the importance of policy implementation.

The impacts of elephants on woodlands have triggered concerns that elephants were converting woodlands to grasslands in African savannas leading to the loss of biodiversity. Therefore, the ob- jectives of the study are to identify the causes, processes and impacts of woodland conversion by elephants and thereby propose a guideline for formulating management strategies. The study is conducted through reviewing published documents on elephant-woodland interaction, factors, mechanisms and processes of woodland conversion to grasslands and impacts on biodiversity have been identified. The study reveals that: 1) the large nutritional and energetic requirements by elephants and high elephant population densities are two driving forces that may cause ele- phants to convert woodlands to grasslands; 2) the process of woodland conversion is not just ele- phant-tree interaction, but usually included other agents such as fires, droughts or other herbi- vores. Woodlands are converted to grasslands or scrublands because elephants, fires, droughts and other herbivores prevent recruitment of tree to larger **class** **sizes** and regeneration of seed- lings of woody plants; 3) whereas there are few studies that support biodiversity decline due to impacts of elephants on woodlands, there are studies that indicate woodland disturbance by ele- phants have benefited and increased diversity of other species. It is concluded that woodland dis- turbance by elephant is not always detrimental to biodiversity, but may create heterogeneity and suitable habitats for other species. In formulating management strategies, considerations must be made to elephant densities, other agents that interact or work with elephants in woodland con- version and the benefits of woodland changes to biodiversity due to elephants.

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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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The importance of a child's early social and academic adjustment to school has been recognised in Britain for some time. Research was conducted at the end of the 1970s on factors influencing successful transition into infant and first school (e.g. Cleave et al, 1982) and nursery school (Blatchford et al, 1982). But a number of factors have led to a renewed interest. Recent initiatives in the U.K. regarding school entry assessments has encouraged interest in more precisely assessing children's adjustment to school, soon after entry. Schools in England enter children in the year within which they are five, and some of these children are only just four years old on entry. There are concerns about the appropriateness of existing teaching methods, **class** **sizes** and staffing. Concerns with behaviour and indiscipline in schools have also heightened awareness of problems posed by some young children in school. There appear to be signs that difficult behaviour in schools is increasing. Day, Tolley, Hadfield, Parking, & Watling (1996) review research linking **class** size with pupil behaviour and argue that large **class** **sizes** are at odds with a wish to improve behaviour in schools and help management of problem behaviour.

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