CHAPTER I INTRODUCTION 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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The study of the subgroups of πΊπΏ(π, πΉ ) in the case when π is infinite dimensional over πΉ has been much more limited and normally requires some additional restrictions. The circumstances here are similar to those present in the early development of Infinite Group Theory. One approach there consisted in the application of finiteness conditions to the study of infinite **groups**. One such restriction that has enjoyed considerable atten- tion in **linear** **groups** is the notion of a finitary **linear** group. In the late 1980βs, R.E. Phillips, J.I. Hall and others studied infinite dimensional lin- ear **groups** under finiteness conditions, namely finitary **linear** **groups** (see [34, 14, 32, 35, 15, 16])). Here πΊ is called finitary if, for each element π β πΊ, the subspace πΆ π (π) has **finite** codimension in π ; the reader is

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Unitary representations. Our primary motivation and model example is the view of infinite dimensional unitary representations into tracial von Neumann algebras as limits of **finite** dimensional unitary representations. By a **finite** dimensional unitary representation (of **degree** r), we mean a homomorphism ΞΊ : F r β U (n) of the free group on r generators into the unitary group U(n). Note that such representations can be given by the r-tuple {ΞΊ(Ξ³ i } r i=1 , where {Ξ³ i } r i=1 are the standard generators of the

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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.

proaches the infimum, the difference π β 2π gets smaller, as required. Returning to the group-theoretic problem stated above (for **degree** π, not necessarily a **prime**), Jordan [10] showed that π΅(π) = β π β 1 + 1 is a lower bound for the minimal **degree**. A substantial improvement of this bound is due to Bochert [3] who showed that π΅(π) = π/8, and if π > 216 then one has an even better bound, namely π΅(π) = π/4. Proofs for the Jordan and Bochert estimates can be found also in Dixon & Mortimor [7], Theorem 3.3D and Theorem 5.4A, respectively. More recently, Liebeck and Saxl [11], using the classification of **finite** simple **groups**, have proved π΅(π) = π/3.

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Over years there has been considerable literature studying global properties of **groups** which are determined by the structure or embedding of their Sylow p-subgroups, where p is a **prime** which is going to be fixed. Most of these results go back to Burnsideβs p-nilpotency criterion stating that a group is p-nilpotent, i.e. it has a normal Hall p β² -subgroup provided that a Sylow p-subgroup is in the centre of its normaliser. As a consequence, a group with cyclic Sylow p-subgroups is p-nilpotent if its order is coprime to p β 1. This result does not remain true for metacyclic Sylow p-subgroups as the alternating group of **degree** 5 shows. However, if the order of a group G is coprime to p 2 β 1 and its

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Indeed for ππ β 0 , this formula also shows an upper bound for πΊπΊ and does not determine the exact number of ππ(πΊπΊ). Also, several results have been verified about conjugacy classes of subgroups of metacyclic ππ -**groups** see [8,9,10]. For example, in [10, Theorem 1.3] it was shown that if πΊπΊ is any **finite** split metacyclic ππ -group for an odd **prime** ππ , that is, πΊπΊ = π»π» β πΎπΎ for subgroups π»π» and πΎπΎ , and if |π»π»| = ππ πΌπΌ and |πΎπΎ| = ππ πΌπΌ +π½π½ , then there exist exactly

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We summarize our notations. cl(a) denotes the conjugacy class of a in G, Ο(G) denotes the set of **prime** numbers dividing the order of G, Ο(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

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. Our Solution. Our solution largely adopts the notion of the pair encoding framework. However, the pair encoding framework cannot properly describe non-**linear** common variables. Therefore, we improves the syntax of pair encoding. The most significant change in our framework is that we decompose variables used as exponents of public keys and master secret keys into two types hidden common variables and shared common variables to express non-linearity in PE schemes as follows:

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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 β(G) has no triangle, then | Ο(G) | β€ 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 β(G) has one triangle and proved that β(G) has at most six vertices and if β(G) has six vertices, then G β P SL(2, 2 f ) Γ A, where A is abelian, | Ο(2 f β Ξ΄) | = 2, and | Ο(2 f + Ξ΄) | = 3 for some Ξ΄ = Β± 1 with f β₯ 10. Furthermore, if β(G) has five vertices, he described all possible cases for such a graph.

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Case 2: t ^ a. (By symmetry, of course, we may also assume that 5 a.) In this case, Q = {Br\G')r\ {B D where r- = (13) or r = (23); also, (5.25) is not necessarily satisfied for all G. We claim that Q may be assumed to be cyclic. To see this, suppose that Q is noncyclic; that is, ( 5 n G) D ( S n Hf^ > Ft. Then S n G*" contains one of F{i,j, 0,0,0,1) or F{i,j, 1,1,1,0) as maximal. Thus, as we saw above, B Cl G^ certainly occurs as the diagonal subgroup of a group in the Ust of Theorem 5.4.1. Since by hypothesis G is in that list, we have by Remark 5.4.2 that {B n Gy = B n G. In particular, B Ci G has a maximal V^4-submodule with centraJiser (a) in B. The faithful **finite** V4-submodules of B with such a maximal submodule were listed in Case 1; of these, we see from the orbit list given in the proof of Proposition 5.2.8 that only the F{i, 1,1,1, 0, 0) and F(0, 0, 0, 0,1,1,1) are normalised by r. For r = (13) and r = (23) in turn, we follow the procedure established in Case 1 (here, w may stiU be chosen as u^) for each G satisfying (5.25)- (5.27) and with one of the aforementioned V4-modules as diagonal subgroup. This wiU yield all **linear** isomorphisms of type I between such G and other **groups** in the Ust of Theorem 5.4.1. Details of these calculations are omitted: no new isomorphisms are obtained, and so from now on we assume that Q is cyclic.

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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 β₯ 2 and we start an induction on n.

The efficiency of direct products of **groups**, stimulated by questions asked by Wiegold in [30], has been studied by several authors; see for example [1], [4], [7], [16]. In this chapter we give general methods for proving that direct products of two or three **groups** possessing **certain** properties are efficient and also give some specific examples. The most general of these examples involve the family of simple **groups** P5L(2, p), for **prime** p > 5. 5L(2, p) is the group of two by two matrices having entries in Zp of determinant one. This group has only one invoΒ lution, the central element, and factoring by the centre yields PSL{2, p). Both of these **groups** are perfect. 5T(2, p) has trivial Schur multiplier and PSL{2, p) has multiplier Cg, its covering group being 5L(2, p).

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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 GaschΛ utz and ItΛ 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. In 2000, Ballester-Bolinches et al. [3] introduced the c-supplementation concept of a **finite** group: A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H β© K β€ H G , where H G = Core G (H) is the

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Having said this however, it should be noted that the nilpotent n-dimensional FpβLie algebra which we associate to each isomorphism class of **groups** of order pn (p > n) whose derived subgroup has exponent dividing p (obtained by ignoring the T βaction) is not, in general, the graded Lie algebra which arises from the filtration of such a group by the lower p-central series. This follows from the fact that for **groups** o f exponent p, the FpβLie algebra we associate is isomorphic to the Lie algebra given by the Campbell-Hausdorff formula and this Lie algebra is uniquely determined by the isomorphism type of the group. This is not the case for the graded construction however, since one always has non-isomorphic **groups** of order pn (p,n > 5) whose Fp βLie algebras arising from the lower p-central series are isomorphic. To see that this is the case, consider the table of **groups** of order p5 given in [12] and in particular the **groups** (in the notation of the paper) ( 15) and </>io(l5)- These are both of exponent p, maximal class 4 and non-isoclinic (hence non-isomorphic), but from the presentations given there one can verify that the 5-dimensional graded Fp βLie algebras arising from their lower central series are both isomorphic to the split extension of the 4-dimensional Abelian Fp βLie algebra by a nilpotent **linear** map o f maximum nilpotency class 4 (for p > 5). By taking direct products of these two **groups** with an elementary Abelian p-group o f the appropriate order one sees that a similar situation holds for any n > 5.

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The paper is organized as follows. Firstly we recall main deο¬nitions concerning **linear** automata over modules. In this account we follow [2]. Then we introduce a special class of **linear** automata, so-called scalar automata. In such automata the module of inner states is equal to the module of letters and transition and output functions are the sums of multiplications by elements of the layer ring. We classify in Theorem 1 the **groups** of scalar automata. The proof is based on the technique presented in [3, Proposition 4.1] and developed in [4, Theorem 4.1] and [5, Proposition 1], where, in fact, **groups** of some scalar automata were calculated. As a corollary, we describe in Theorem 2 **groups** of **linear** automata over a ο¬nite ο¬eld whose space of states is equal to this ο¬eld. These results may be regarded as a contribution to the theory of self-similar **groups** ([6]). 2. Let R be a commutative ring with unit, R β

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Theorem 3. Let G be a group generated by (**finite**) automaton (with no cycles with exit) A = hX, Q, Ο, Ξ»i over an alphabet X, P < S(X) and for every state q β Q the permutation Ξ»(q, Β·) belongs to the group P. Then the group P β G is generated by (**finite**) automaton (with no cycles with exit) over an alphabet X.

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I n t h i s t h e s i s , we i n v e s t i g a t e t h e d e f i c i e n c y o f t h e **groups** P S L ( 2 , p ^ ) . J .A . Todd gave p r e s e n t a t i o n s f o r PSL(2,p^) which use l a r g e numbers o f g e n e r a t o r s and r e l a t i o n s ("A second n o t e on t h e l i n e a r f r a c t i o n a l g r o u p . " J . London Math, Soc. 2 (1936) 103-107). S t a r t i n g w i t h t h e s e , we o b t a i n , a t b e s t , d e f i c i e n c y -1 p r e s e n t a t Β io n s f o r PSL(2,2^) (5 S L ( 2 ,2 ^ )) and d e f i c i e n c y -6 p r e s e n t a t i o n s f o r P S L (2 ,p ^ ) , p an odd p r im e . I f p^ e -l(m od 4 ) , t h e l a t t e r can

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varieties of metabelian **groups**). The basic (but in its full generality entirely h o p e l e s s ) problem in this theory is to describe all metabelian varieties and the lattice lat(AA) they form, and indeed most of the results obtained so far concern aspects of this problera.

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