In studying finite linear groups of fixed degree over the complex field, it is convenient to restrict attention to irreducible, unimodular, and quasiprimitive grou[r]

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T h e second case we consider is
G = (axj^,, F ( i , i , k, 0 , 0 , 0 , - 1 ) ) .
B y inspection of the orbit list in the proof of Proposition 5.2.8, we see that all six elements of S3 are relevant here. When j , k, I are pairwise different, the six S3- conjugates of G differ even in their intersections with B, but at that level only by permutations of the three relevant parameters, and precisely one of them has those parameters in increasing order. Passing to the unique 5V^-conjugates in the list of T h e o r e m 5.3.11 has no effect on those three parameters. Hence, there we find six distinct **groups**, all in the third line (and with = 0), precisely one with the three parameters in increasing order. It is not necessary to calculate how the values of the parameters 7 , /i of the six **groups** depend on the parameter values of G , since it follows from what has been said so far that precisely one of them is in the list of T h e o r e m 5.4.1.

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1. Introduction
Throughout this note, G denotes a **finite** group. The relationship between the proper- ties of the Sylow subgroups of a group G and its structure has been investigated by many authors. Starting from Gasch˝ utz and It˝ o ([10], Satz 5.7, p.436) who proved that a group G is solvable if all its minimal subgroups are normal. In 1970, Buckely [4] proved that a group of odd order is supersolvable if all its minimal subgroups are normal (a subgroup of **prime** order is called a minimal subgroup). Recall that a subgroup is said to be S-permutable in G if it permutes with all Sylow subgroup of G. This concept, as a generalization of normality, was introduced by Kegel [11] in 1962 and has been studied extensively in many notes. For example, Srinivasan [15] in 1980 obtained the supersolvability of G under the assumption that the maximal subgroups of all Sylow subgroups are S-permutable in G.

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In mathematics, there is a branch related with the study of uncertainty named probability. This probability can be applied to another field of mathematics such as group theory. The results obtained are very interesting since the calculation dealt with regular properties of elements of a **certain** set.

University of Warwick institutional repository: http://go.warwick.ac.uk/wrap
A Thesis Submitted for the **Degree** of PhD at the University of Warwick
http://go.warwick.ac.uk/wrap/63211
This thesis is made available online and is protected by original copyright.

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We summarize our notations. cl(a) denotes the conjugacy class of a in G, π(G) denotes the set of **prime** numbers dividing the order of G, ϕ(n) denotes the Euler function that counts the positive integers less than n that are relatively **prime** to n, F(G) denotes the subgroup generated by all normal nilpotent subgroups of G, O p (G) denotes the unique maximal normal p-subgroup of G, F p,q denotes

Proof. Let n be the number of the not necessarily distinct **prime** divisors of |G|. Let us denote the **prime** |A| by p. The factorization G = AB implies that |G| = |A||B|. In the special case n = 1 it follows that |G| = |A| = p and |B | = 1. Therefore G = A and B = {e}. Plainly hB i 6= G and so for the special case n = 1 the theorem is proved. For the remaining part of the proof we assume that n ≥ 2 and we start an induction on n.

Abstract
Motivated by the theory of graph limits, we introduce and study the convergence and limits of **linear** representations of **finite** **groups** over fi- nite fields. The limit objects are infinite dimensional representations of free **groups** in continuous algebras. We show that under a **certain** inte- grality condition, the algebras above are skew fields. Our main result is the extension of Schramm’s characterization of hyperfiniteness to **linear** representations.

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Therefore consider a case when K is **finite**. Being **finite**, K is soluble.
Then D = [K, K ] 6= K. If we suppose that a set Π(K/D) contains two different primes, then K is a product of two proper normal subgroups. In this case K has **finite** central dimension, as shows Lemma 1. Thus K/D is a p-group for some **prime** p. If we suppose that K/D is not cyclic, then again K is a product of two proper normal subgroups, which follows that K has **finite** central dimension. This contradiction shows that K/D is a cyclic p-subgroup. Let x be an element of K such that K/D = hxDi.

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In Chapter 1, we briefly recall a few definitions and well-known results from several relevant topics, which constitute the minimum prerequisites for the subsequent chapters. In this chapter, we also fix **certain** notations. Given a subgroup K of a group G and an element x ∈ G, we write C K (x) and C ` K ( x) to denote the sets {k ∈ K : kx = xk} and {kxk −1 ∈ G : k ∈ K}

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Also, several results have been verified about conjugacy classes of subgroups of metacyclic 𝑝𝑝 -**groups** see [8,9,10].
For example, in [10, Theorem 1.3] it was shown that if 𝐺𝐺 is any **finite** split metacyclic 𝑝𝑝 -group for an odd **prime** 𝑝𝑝 , that is, 𝐺𝐺 = 𝐻𝐻 ⋉ 𝐾𝐾 for subgroups 𝐻𝐻 and 𝐾𝐾 , and if |𝐻𝐻| = 𝑝𝑝 𝛼𝛼 and |𝐾𝐾| = 𝑝𝑝 𝛼𝛼 +𝛽𝛽 , then there exist exactly

Thus 𝐺 is finitary if and only if centdim 𝐹 ⟨𝑔⟩ is **finite** for every 𝑔 ∈ 𝐺. At this point, it is worth remarking that the above notion of central dimen- sion of a **linear** group does not define a class of **groups**, since it heavily relies in the way in which the **linear** group 𝐺 is embedded in a particular general **linear** group. In fact, given an abstract group 𝐺, it is easy to construct embeddings of 𝐺 in the same general **linear** group such that 𝐺 has infinite or **finite** central dimension depending on the embedding.

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g 3 g 4 e g 1 g 2
g 4 e g 1 g 2 g 3
If we had chosen any non trivial object, , g 1 , at the beginning, we would end up with the same table. All we would be doing is changing the symbols with respect to positions, but the structure is the same; we chose g 1 without loss of generality. Then, we had to choose a second object for g 1 ∗ g 1 ; the only condition is that g 2 , e, g 1 . Then we have to select an object g 3 with the only restriction it be di ff erent than , e, g 1 , g 2 so we took g 3 = g 1 ∗ g 2 . Finally, we had to choose an object for g 1 ∗ g 3 and it had to be di ff erent than e, g 1 , g 2 , g 3 so we trivially choose g 4 . No other restrictions were imposed during this process so that it does not depend on our initial choice, nor the second, third, etc. We simply have to take new elements, without observing any other relations. The group is defined solely by the number of objects. In the next examples, we will have to satisfy **certain** restrictions on each step of our process, to find the canonical representation and naming. The numerical table for this group is given by the naming function, ρ : e = 4, g 1 = 3, g 2 = 2, g 3 = 1, g 4 = 0, where all the objects are equivalent. This means we can take other canonical naming functions, for example, ρ : e = 4, g 4 = 3, g 3 = 2, g 2 = 1, g 1 = 0. The numeric table is

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stricter structural restrictions. They defines the scheme satisfying those restrictions as a regular encoding. For example, h i h i 0 cannot be used and computed in their framework. The work of Kim et al. [16, 1] also explicitly defines linearity in common variables of keys and ciphertexts as a new property for their security analysis. Also, the techniques suggested in [11, 1] assume that linearity in common variables and use them for their proofs, implicitly using the structural definition of pair encodings. As described in the overview of pair encoding, the pair encoding was not defined only by properties, but also required to have a **certain** structure which is **linear** in common variables.

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information see the survey paper [2].) In general, it seems that the **prime** **degree** graphs contain many edges and thus they should have many triangles, so one of the cases that would be interesting is to consider those **finite** **groups** whose **prime** **degree** graphs have a small number of triangles. In [4], the author studied **finite** **groups** whose **prime** **degree** graphs have no triangle. In particular, he proved that if ∆(G) has no triangle, then | ρ(G) | ≤ 5. He also obtained a complete classification of all **finite** **groups** whose **prime** **degree** graphs contain no triangle with five vertices. In [5], the author studied **finite** **groups** whose **prime** **degree** graphs have at most two triangles. In particular, in [5, Theorem A], he considered the case where ∆(G) has one triangle and proved that ∆(G) has at most six vertices and if ∆(G) has six vertices, then G ≃ P SL(2, 2 f ) × A, where A is abelian, | π(2 f − δ) | = 2, and | π(2 f + δ) | = 3 for some δ = ± 1 with f ≥ 10. Furthermore, if ∆(G) has five vertices, he described all possible cases for such a graph.

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We shall prove that the structure of this group W as a vector space is isomorphic to the Kronecker product of t two dimensional.. vector spaces and that the repre[r]

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is an infinitely differentiable action of a compact group G on a real manifold X then every real valued function invariant with respect to G possesses extrema on each stratum correspondi[r]

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On the other hand, in chapter 6 we tise the Lie ring functors to solve a restricted form of a conjecture of J. Moody [22] by exhibiting, for a **prime** p greater than or equal to the positive integer n, a natural, but not functorial, one-to-one correspondence between iso
morphism classes of **finite** **groups** of order p" whose derived subgroup has exponent dividing p, and isomorphism classes of nilpotent Fp[T]/(T" )—Lie algebras L o f Fp—dimension n in which T[L,L] = 0. By viewing such an algebra as a nilpotent Fp —Lie algebra equipped with a nilpotent element of its centroid one obtains a “formula” for the number of such **groups**. This applies, in particular, to the **groups** o f order p7 since the 7-dimensional nilpo

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Obviously all elements of p-**groups** have **prime** power orders. So also in arbitrary **groups** we want to consider sets of **prime** power order generators.
In this purpose we introduce the concept of a pp-element which simplifies our considerations. So we say that an element g ∈ G is a pp-element if it has **prime** power order, while by p-element, as usual, we mean an element of order being a power of a **prime** p. Many authors have studied similar problems concerning sets of not only pp-generators, see for instance [1, 8, 9, 11] and the reference therein. In particular in [1] **groups** in which all minimal generating sets have the same size are classified.

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If n is an integer, then we denote by π(n) the set of all **prime** divisors of n. If G is a **finite** group, then π(|G|) is denoted by π(G). Also the set of orders of elements of G is denoted by ω(G). This set is closed under divisibility relation. The **prime** graph Γ(G) of a **finite** group G is defined as a graph with vertex set π(G) in which two distinct primes p, p 0 ∈ π(G) are adjacent iff G contains an element of order pp 0 . Denote by t(G) the maximal number of primes in π(G) pairwise nonadjacent in Γ(G). Also we denote by t(p, G) the maximal number of vertices in the independent sets of Γ(G) containing p.

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