Semidirect Products Of Cyclic Groups Research Articles

On BCH split metacyclic codes

<p style='text-indent:20px;'>Recently, Borello and Jamous have investigated some lower bounds on the dimension and minimum distance for dihedral codes, in analogy with the theory of BCH codes. In this paper, we extend some of their results to split metacyclic codes, that is, codes over semidirect products of cyclic groups.</p>

Commuting Graphs of Split Metacyclic Groups

We study the link between groups and graphs created by considering the commuting graph of a group. We focus our efforts on groups that can be represented as the semidirect product of cyclic groups, and we describe the commuting graphs of two classes of such groups.

A remark on a conjecture of Derksen

Let V be a complex representation of a finite group G of order g. Derksen conjectured that the pth syzygies of the invariant ring Sym(V ) are generated in degrees ≤ (p+ 1)g. We point out that a simple application of the theory of twisted commutative algebras — using an idea due to Weyl — gives the weaker bound pg, almost for free. Fix a finite group G of order g. Let V be a finite dimensional complex representation of G and put R = Sym(V )G, which we regard as a graded ring. Let E ⊂ R be a homogeneous vector subspace generating R as an algebra, and put S = Sym(E), so that S → R is a surjection of graded algebras. The space Torp (R,C) is then a graded vector space, called the space of p-syzygies of R. We let sp(V ;E) be the maximal degree occurring in it. One can show that if E ⊂ E′ then sp(V ;E) ≤ sp(V ;E′). Furthermore, sp(V ;E) is independent of E if E is chosen to be minimal; we denote this common value by sp(V ). We then have the following conjecture of Derksen (see [D, Conj. 3] for a more precise version): Conjecture 1. We have sp(V ) ≤ (p+ 1)g for any V . Derksen [D, Thm. 2] proved the conjecture for p = 1, but the general case is open. To state our main result, we first introduce some notation. Let β(V ) be the minimal integer such that R is generated in degrees ≤ β(V ). Let β be the maximum value of β(V ) over all V , the so-called Noether number of G. Noether’s theorem [W, §3] states that β ≤ g. Let d1, . . . , dn be the degrees of the irreducible representations of G and let m be the sum of the di. Finally, put δp = (β − 1)g − (m− 1)βp; note that this is negative for p 0. We then have: Theorem 2. We have sp(V ) ≤ β2mp+ δp for any V . Although this is weaker than the conjecture, the bound is significant since it is independent of V . Using the fact that β and m are both bounded by g, we deduce the following corollary: Corollary 3. We have sp(V ) ≤ pg3 for any V . Remark 4. In fact, m and β are often strictly smaller than g, sometimes significantly so. We have m ≤ √ng, where n is the number of conjugacy classes in G. Typically, n is much smaller than g; for example, if G is a symmetric group then n = O(g ), for any > 0. The currently known bounds on β are not very sharp. A result of Cziszter–Domokos [CD] states that β < 1 2g unless G has a cyclic subgroup of index two, or is one of four exceptions. A conjecture of Pawale [W, Conj. 3.9] asserts that if G is the semi-direct product of cyclic groups of prime orders p and q, with q | p− 1, then β = p+ q − 1; note that m = p+ q − 1 in this case as well. When p and q are approximately equal, this gives m = β = O( √ g). Thus, in many case, our theorem gives a bound of the form sp(V ) = O(pg θ) with θ < 3. Remark 5. Theorem 2 (and our proof of it) is valid over any field of characteristic zero. Derksen also proposed his conjecture in positive characteristic not dividing g. It would be interesting if our proof could be adapted to work in this setting. We now prove the theorem. Put sp(V ) = sp(V ;E) where E = ⊕β i=1Ri. As discussed, we have sp(V ) ≤ sp(V ), so it suffices to bound the latter. The key result is the following lemma. Date: November 2, 2012. The author was supported by NSF fellowship DMS-0902661. 1

Mapping outSU(5)grand unified theories with non-Abelian discrete flavor symmetries

We construct a class of supersymmetric $SU(5)$ grand unified theory (GUT) models that produce nearly tribimaximal lepton mixing, the observed quark mixing matrix, and the quark and lepton masses, from discrete non-Abelian flavor symmetries. The $SU(5)$ GUTs are formulated on five-dimensional throats in the flat limit and the neutrino masses become small due to the type-I seesaw mechanism. The discrete non-Abelian flavor symmetries are given by semidirect products of cyclic groups that are broken at the infrared branes at the tip of the throats. As a result, we obtain $SU(5)$ GUTs that provide a combined description of non-Abelian flavor symmetries and quark-lepton complementarity.

Further results on the enumeration of hamilton paths in Cayley digraphs on semidirect products of cyclic groups

First, let m and n be positive integers such that n is odd and gcd ( m , n ) = 1 . Let G be the semidirect product of cyclic groups given by G = Z 8 m ⋊ Z 2 n = 〈 x , y : x 8 m = 1 , y 2 n = 1 , and yxy - 1 = x 4 m + 1 〉 . Then the number of hamilton paths in Cay ( G : x , y ) (with initial vertex 1) is one fewer than the number of visible lattice points that lie on the closed quadrilateral whose vertices in consecutive order are ( 0 , 0 ) , ( 4 mn 2 + 2 n , 16 m 2 n ) , ( n , 4 m ) , and ( 0 , 8 m ) . Second, let m and n be positive integers such that n is odd. Let G be the semidirect product of cyclic groups given by G = Z 4 m ⋊ Z 2 n = 〈 x , y : x 4 m = 1 , y 2 n = 1 , and yxy - 1 = x 2 m - 1 〉 . Then the number of hamilton paths in Cay ( G : x , y ) (with initial vertex 1) is ( 3 m - 1 ) n + m ⌊ ( n + 1 ) / 3 ⌋ + 1 .

Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups

For G any finite group, the left and right regular representations λ, respectively ρ of G into Perm ( G ) map G into InHol ( G ) = ρ ( G ) ⋅ Inn ( G ) . We determine regular embeddings of G into InHol ( G ) modulo equivalence by conjugation in Hol ( G ) by automorphisms of G, for groups G that are semidirect products G = Z h ⋊ Z k of cyclic groups and have trivial centers. If h is the power of an odd prime p, then the number of equivalence classes of regular embeddings of G into InHol ( G ) is equal to twice the number of fixed-point free endomorphisms of G, and we determine that number. Each equivalence class of regular embeddings determines a Hopf Galois structure on a Galois extension of fields L / K with Galois group G. We show that if H 1 is the Hopf algebra that gives the standard non-classical Hopf Galois structure on L / K , then H 1 gives a different Hopf Galois structure on L / K for each fixed-point free endomorphism of G.

Quantum algorithms for the hidden subgroup problem on some semi-direct product groups by reduction to Abelian cases

We consider the hidden subgroup problem on the semi-direct product of cyclic groups Z N ⋊ Z p , where p is a prime that does not divide p j − 1 for any of the prime factors p j of N, and show that the hidden subgroup problem can be reduced to other ones for which solutions are already known.

Active Sums I

Given a generating family F of subgroups of a group G, closed under conjugation and with partial order compatible with inclusion, a new group S can be constructed, taking into account the multiplication in the subgroups and their mutual actions given by conjugation. The group S is called the active sum of F, has G as a homomorph and is such that S/Z(S) ' G/Z(G), where Z denotes the center. The basic question we investigate in this paper is: when is the active sum S of the family F isomorphic to the group G? The conditions found to answer this question are often of a homological nature. We show that the following groups are active sums of cyclic subgroups: free groups, semidirect products of cyclic groups, Coxeter groups, Wirtinger approximations, groups of order p3 with p an odd prime, simple groups with trivial Schur multiplier, and special linear groups SLn(q) with a few exceptions. We show as well that every finite group G such that G/G0 is not cyclic is the active sum of proper normal subgroups.

Noether's bound for polynomial invariants of finite groups

Let G be a finite group acting linearly on the polynomial algebra \(\Bbb C [V]\). We prove that if G is the semi-direct product of cyclic groups of odd prime order, then the algebra of polynomial invariants is generated by its elements whose degree is bounded by \({5 \over 8}|G|\). As a consequence we derive that \(\Bbb C [V]^G\) is generated by elements of degree \(\leqq {3 \over 4}|G|\) for any non-cyclic group G. This sharpens the improved bound for Noether's Theorem due to Schmid.

Split Metacyclic p-Groups That Are A-E Groups

A group G is an A-E group if the endomorphism nearring of G generated by its automorphisms equals the endomorphism nearring generated by its endomorphisms. In this paper we set out to determine those p-groups G that are semidirect products of cyclic groups and are A-E groups. We show that no such groups exist when p = 2. When p is odd, we show that G is an A-E group whenever the nilpotency class of G is less than p. Examples are given to show no conclusion can be drawn when the nilpotency class is greater than or equal to p.

Morava K-theory of classifying spaces: Some calculations

We develop some basic methods for calculating Morava K-theories of compact Lie groups, and compute certain pivotal examples. We show that K ̃ (2) ∗BP has odd elements, where P is the 3-Sylow subgroup of GL 4 (Z/3). This disproves a conjecture of Hopkins, Kuhn and Ravenel. We also calculate Morava K-theories of semidirect products of cyclic groups with elementary abelian groups, and prove a new theorem on complex-oriented cohomology of BO(K).

A Note on Constructing Large Cayley Graphs of Given Degree and Diameter by Voltage Assignments

Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree $\le 15$ and diameter $\le 10$ have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. This opens up a new possible direction in the search for large vertex-transitive graphs of given degree and diameter.

The semidirect products of finite cyclic groups that areI-E groups

AnI-E group is a group in which the endomorphism near-ring generated by the group's inner automorphisms equals the endomorphism near-ring generated by its endomorphisms. In this paper we shall completely determine the finite groups that are semidirect products of cyclic groups and areI-E groups.

Semidirect products of 𝐼-𝐸 groups

An I-E group is a group G in which the endomorphism near-ring generated by the inner automorphisms of G equals the endomorphism near-ring generated by the endomorphisms of G. In this paper we obtain a result characterizing when a semidirect product of I-E groups of relatively prime orders is an I-E group. We then use this result to show that a semidirect product of cyclic groups of relatively prime orders is an I-E group.

New results for the degree/diameter problem

AbstractThe results of computer searches for large graphs with given (small) degree and diameter are presented. The new graphs are Cayley graphs of semidirect products of cyclic groups and related groups. One fundamental use of our “dense graphs” is in the design of efficient communication network topologies. © 1994 by John Wiley &amp; Sons, Inc.

Connected Cayley graphs of semi-direct products of cyclic groups of prime order by abelian groups are hamiltonian

In this paper it is shown that every connected Cayley graph of a semi-direct product of a cyclic group of prime order by an abelian group is hamiltonian. In particular, every connected Cayley graph of a group G is hamiltonian provided that G is of order greater than 2 and it contains a normal cyclic subgroup N of prime order such that the quotient group G/ N is abelian and its order is relatively prime to that of N.

Construction of the symmetry groups of polymer molecules

A derivation of the line groups, which are the symmetry groups of the stereo-regular polymer molecules, is presented. Every line group is an extension of a one-dimensional translation group by a point group. The point group is either cyclic or a semi-direct product of cyclic groups. The expounded method of derivation of the line groups consists of first extending by the cyclic point groups and then obtaining the rest of the line groups by unification within the Euclidean group. To this purpose, a simple test is given to decide whether two line groups multiply into a third one. The method displays the subgroup structure of the line groups, relevant to the construction of their irreducible representations. All the line groups are derived and tabulated.

Crystal symmetry groups and their representations I lattice extensions of point groups in R 2 and R 3

We classify all point groups in R 2 and R 3 according to their decomposition into m-fold semidirect products of cyclic groups. This product is characterized completely by means of m−1 matrices α (1),…, α ( m−1) with non-negative integer elements. For any arithmetic class [ P n i ] in R n ( n=2, 3) we define the set D( P n i , T) of representative translations with the following properties: (a) The elements of D( P n i , T) are in one-to-one correspondence with defining relations of P n i ; (b) D( P n i , T) determines uniquely all equivalence classes of the factors of P n i over T. We present: (i) a construction of all sets D( P n i , T) in R 2 and R 3; (ii) a construction of all algebraic classes of space groups and a derivation of all group operations which transform one algebraic class into another; (iii) a derivation of formulas expressing the multiplication table of any space group by means of the representative translations and matrices α (1),…, α ( m−1) . The results (i), (ii) and (iii) are obtained without making use of the notion of non-primitive translations.