Lets discuss some results for groups with minimality condition. Old, recent and new results on quasinormal subgroups. By closing this message, you are consenting to our use of cookies. Example of nonlinearizable quasicyclic subgroup of automorphism group of polynomial algebra article pdf available january 2015 with 19 reads how we measure reads. As we will see in the next section, dihedral codes, as all group codes, form a subfamily of quasicyclic codes. The quasicyclic group is defined in the following equivalent ways. Characters and quasipermutation representations 1823 now it is easy to see that z,y. Kur60 implies that the p group asplits into the direct sum of its nite cyclic and in nite quasi cyclic subgroups groups. Sep 11, 20 analogous to 1generator quasi cyclic codes, we investigate the structural characterization and enumeration of 1generator quasi abelian codes. For, if p is a prime, then any cyclic group cpn extended by any cyclic group cpm has all subgroups quasinormal provided, when p 2 and n 2, the cyclic subgroup of order 4 in c2n is central in the extension. Cyclic groups september 17, 2010 theorem 1 let gbe an in nite cyclic group. If p is a prime, then the quasicyclic group of type p, denoted cp.
To learn about our use of cookies and how you can manage your cookie settings, please see our cookie policy. Construction of quasicyclic codes by thomas aaron gulliver b. Finite fields and their applications, elsevier, 2012, 18 5, pp. We need the following proposition in the proof of theorem 1.
It is the quotient where is the group of all rational numbers that can be expressed with denominator a power of. The cyclic subgroup intersection graph of a nite group g, csig, is a simple graph with nontrivial cyclic subgroups as vertex set. However, most pgroups are of class 2, in the sense that as n. Pdf on the algebraic structure of quasicyclic codes.
Morgan barbier, christophe chabot, guillaume quintin. The additive group q of rational numbers and the quasicyclic pgroup z p. Pdf by a quasipermutation matrix we mean a square matrix over the complex field with nonnegative integral trace. Proposition 1 reduces the problem of the structure of irreducible sylow p subgroups of gln, t to the classification of irreducible primitive p subgroups. Old, recent and new results on quasinormal subgroups 129 actually normalises a 17.
As g has an abelian maximal subgroup, these must have degree p. The prufer pgroup is the unique infinite pgroup which is locally cyclic every finite set of elements generates a cyclic group. We investigate the class of quasipotent, cyclic subgroup separable groups, producing many examples and showing how it behaves with respect to certain group constructions. A necessary condition for gto have primitive relations is that pmust be of the form pdkia, where aacts faithfully on c, and in particular is cyclic of order pm, say, adhhi.
Quasipermutation representations of groups of order 64 houshang behravesh and ghodrat gha. A subgroup h of g is called 7t quasi normal in g if h permutes with every sylow p subgroup of g for every p in n. In the next proposition, we prove that a ring whose group of units is cyclic or quasicyclic must have characteristic 0,2,4,q or 2q, where q is a fermat prime. The cyclic group of order two lies in the family of cyclic groups, of general linear groups, and of symmetric groups.
Sylow psubgroups of the general linear group over a skew. Construction of quasicyclic selfdual codes sunghyu han, jonlark kimy, heisook lee z, and yoonjin lee x december 17, 2011 abstract there is a onetoone correspondence between quasi cyclic codes over a nite eld. Pdf quasipermutation representations of metacyclic p. Finite nilpotent groups whose cyclic subgroups are 1579 theorem 2. Finite fields article pdf available in ieee transactions on information theory 477. Furthermore, for every positive integer n, nz is the unique subgroup of z of index n. Quasipotency and cyclic subgroup separability jose burilloa. In 1957, abhyankar made a conjecture ab1 concerning the fundamental group of an a. The latter group may be realized as the union of all cyclic groups of order a power of p. This can be done on the basis of the following result from the theory of finite p groups.
The number of homomorphic images of an abelian group greg oman ohio university, 321 morton hall athens, oh 45701, usa. Let g be a finite group and let n be a set of primes. If 1, meaning that a0 a00, then the old result of kulikov see book kur60 implies that the pgroup asplits into the direct sum of its nite cyclic and in nite quasicyclic subgroups groups. The prufer p groups are the only infinite abelian groups with this property. In fact it is easy to prove that, when p is an odd prime, then cgqg. Notes on tt quasi normal subgroups in finite groups ren yongcai communicated by warren j. Note on the cyclic subgroup intersection graph of a nite group. Recalling that f p is a cyclic group of order and exponent both equal to p 1, our motivating questions throughout the paper will be. But avoid asking for help, clarification, or responding to other answers. Faustova, noninvariance of finite automorphism groups for indecomposable rank2 groups by a quasiisomorphism, ableian groups and modules, tomsk.
Autcis either isomorphic to d8or has normal prank one. The essential idea is to use a cyclic group of automorphisms of the code to represent it as a module over the polynomial ring fx. We begin with properties we have already encountered in the homework problems. Minimal normal subgroups of a finite group mathoverflow. We show that the class of binary selfdual strictly quasi abelian codes is asymptotically good. Analogous to 1generator quasicyclic codes, we investigate the structural characterization and enumeration of 1generator quasiabelian codes. Thus, the corollary also describes the finite simple groups in which a sylow 2subgroup has an abelian subgroup of index 2. Quasipermutation representations of groups of order 64. We present the first explicit construction of a binary symmetric code with constant rate and constant distance i. Thus there is an easy characterization of pgroups of class 1.
In this paper we will calculate cg, qg, and p g, where g is a metacyclic p group with non cyclic center and p is either 2 or an odd prime number. It is the direct limit of the chain of groups where the maps are multiplication by. Primitive groups can be studied by the way the group acts on its socle in many cases, indeed it is common in those cases to view the socle as the. Finite nilpotent groups whose cyclic subgroups are ti. Noethers problem for some pgroups shoujen hu mingchang kang. A subgroup h of g is called 7tquasinormal in g if h permutes with every sylow psubgroup of g for every p in n. Cyclic elastoplastic fracture diagram and some parameters of cyclic crack growth resistance for the plastic steels l. It is the group, under multiplication, of all complex roots of unity for all. A group is pquasielementary for a prime p if has a cyclic normal subgroup of order prime to pso that the quotient is a pgroup. We show that the class of binary selfdual strictly quasiabelian codes is asymptotically good. Finite group with cyclic sylow psubgroup mathematics stack. If h is a subgroup of g and n is a normal subgroup of g, then h and gn are quasicorep pgroups. Symmetry free fulltext on finite quasicorep pgroups.
As g has cyclic center, it has faithful irreducible representations. Subgroups of pquasielementary groups are pquasielementary. Let gdcipbe quasielementary, where cis cyclic of prime order qand pis a pgroup with p. In the former situation, it is shown in 21 that the hypothesis that g is. On quasi cyclic codes as a generalizat ion of cyclic codes. Inthis paper we adopt the same approach to provide new insight into the algebraic structure of quasicyclic codes. Under this correspondence, a cyclic or quasi cyclic summand z p is replaced by an equivalence class of size. Reduction of ideals relative to an artinian module and the dual of burchs inequality. Thanks for contributing an answer to mathematics stack exchange. The group a is an abelian group constructed from a quasicyclic group, and by the minp condition every p. One proof of the sylow theorems exploits the notion of group action in various. A, where w is dihedral or semidihedral and a is cyclic of order 2 or a fourgroup, were considered by harada 9, gorenstein and harada 7, and smith 10.
On quasi cyclic codes as a generalization of cyclic codes morgan barbier, christophe chabot, guillaume quintin to cite this version. An extra special group gis a pgroup whose center zis cyclic of order psuch that gzis an elementary abelian pgroup nice treatments of extra special groups. Notice, however, that there is a largest possible class for each order. Assume that s has a direct factor d which is either nonabelian dihedral or quasi dihedral. The same is true if cpn is replaced by any abelian pgroup h of. Cyclic elastoplastic fracture diagram and some parameters. Abhyankars local conjecture on fundamental groups abstract. In this paper, we investigate the structure of finite quasicorep pgroups. Hence g is isomorphic to a subgroup of gl p, f, where f is the complex numbers, for instance. Dihedral or quasidihedral direct factors of sylow 2subgroup. Then g is quasicorep if and only if hxpie g, for any element x in g. Quasipermutation representations of metacyclic pgroups with noncyclic. Pdf example of nonlinearizable quasicyclic subgroup of. Note on the cyclic subgroup intersection graph of a nite group elaheh haghi and ali r.
Given an elliptic curve eq, how often is the group ef p cyclic. On quasicyclic codes as a generalization of cyclic codes. Let g be a finite p group which has all its abelian normal divisors cyclic. A in a has only a finite number of quasicyclic groups. The number of homomorphic images of an abelian group. Let g be a nonabelian finite p group with an abelian maximal subgroup m and cyclic center. Also here we will state some results without using of 2.
A structure theorem for product sets in extra special groups. All cancellable abelian groups described in corollary 2. However, in special cases they can exert more control. Gis isomorphic to z, and in fact there are two such isomorphisms. It turns out that allowing g to be arbitrary, results separate naturally into the two cases where the cyclic subgroup a is. If p 3, we prove that every quasi core3 3 group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. Faustova, noninvariance of finite automorphism groups for indecomposable rank2 groups by a quasi isomorphism, ableian groups and modules, tomsk. Notes on ttquasinormal subgroups in finite groups ren yongcai communicated by warren j. To illustrate the use of our algorithm, we give an explicit expression of the structure of h0x,l when xis a cyclic pgroup. Pdf by a quasi permutation matrix we mean a square matrix over the complex field with nonnegative integral trace. This follows from the structure of dihedral groups and quasi dihedral groups.
In this paper we will calculate cg, qg, and pg, where g is a metacyclic pgroup with noncyclic center and p is either 2 or an odd prime number. Edge transitive ramanujan graphs and symmetric ldpc good. On groups with the weak wide commensurable property. Rocky mountain journal of mathematics project euclid. Thus, each group of ulm type 1 can be naturally associated with an equivalence structure. Also here we will state some results without using of. Quasi permutation representations of metacyclic pgroups with non cyclic. Finite nilpotent groups whose cyclic subgroups are tisubgroups. Saksonov for helpful discussions on a number of questions arising in connection with this article. G be the semidirect product of a quasip group q with a. Moreover, the code is ldpc and its constraint space is generated by the orbit of one constant weight constraint under the group action. Homomorphic images of an abelian group 111 that p is prime and pg g. Noethers problem for some pgroups new york university. Characters and quasipermutation representations of.
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