# Top PDF On certain finite linear groups of prime degree

### On certain finite linear groups of prime degree

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]

### The structure of infinite dimensional linear groups satisfying certain finiteness conditions

The study of the subgroups of πΊπΏ(π, πΉ ) in the case when π is infinite dimensional over πΉ 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 1980βs, 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 πΊ is called finitary if, for each element π β πΊ, the subspace πΆ π (π) has finite codimension in π ; the reader is

### Convergence and limits of linear representations of finite groups

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

### The Multiplicative Degree of Some Finite Groups

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.

### On Galois groups of prime degree polynomials with complex roots

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.

### On metacyclic subgroups of finite groups

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 Burnsideβs p-nilpotency criterion stating that a group is p-nilpotent, i.e. it has a normal Hall p β² -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

### The precise value of commutativity degree in some finite groups

Indeed for ππ β  0 , this formula also shows an upper bound for πΊπΊ and does not determine the exact number of ππ(πΊπΊ). Also, several results have been verified about conjugacy classes of subgroups of metacyclic ππ -groups see [8,9,10]. For example, in [10, Theorem 1.3] it was shown that if πΊπΊ is any finite split metacyclic ππ -group for an odd prime ππ , that is, πΊπΊ = π»π» β πΎπΎ for subgroups π»π» and πΎπΎ , and if |π»π»| = ππ πΌπΌ and |πΎπΎ| = ππ πΌπΌ +π½π½ , then there exist exactly

### On finite groups having a certain number of cyclic subgroups

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

### A New Encoding Framework for Predicate Encryption with Non-Linear Structures in Prime Order Groups

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:

### On nonsolvable groups whose prime degree graphs have four vertices and one triangle

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.

### Finite irreducible linear 2-groups of degree 4

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.

### Certain finite abelian groups with the Redei $k$-property

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.

### On the efficiency of finite groups

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).

### Finite Groups with Certain Permutability Criteria

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

### Prime power lie algebras and finite p groups

Having said this however, it should be noted that the nilpotent n-dimensional FpβLie 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 βaction) 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 FpβLie 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 βLie 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 βLie algebras arising from their lower central series are both isomorphic to the split extension of the 4-dimensional Abelian Fp βLie 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.

### Groups of linear automata

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

### Finite groups as groups of automata with no cycles with exit

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.

### On some invariants of finite groups

Groups G with Ο|G| = 1 are known as p-groups and are extensively studied, with some specific methods (see [2] and subsequent volumes of this monograph). They have many interesting properties. For example, they are nilpotent groups and every nilpotent group is a direct product of p-groups with coprime orders.

### Presentations of linear groups

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