Group Of Order 2n Research Articles

On the length of the group algebra of the dihedral group in the semi-simple case

The length of group algebras for the dihedral groups is calculated in the semi-simple case. For arbitrary n > 2, we prove that the length of the group algebra of the dihedral group of order 2n over an arbitrary field of a characteristic not dividing 2n is equal to n.

Some structural and closure properties of an extension of the q-tensor product of groups,

In this work we study some structural properties of the group q a non-negative integer, which is an extension of the q-tensor product where G and H are normal subgroups of some group L. We establish by simple arguments some closure properties of when G and H belong to certain Schur classes. This extends similar results concerning the case q = 0 found in the literature. Restricting our considerations to the case G = H, we compute the q-tensor square for q odd, where Dn denotes the dihedral group of order 2n. Upper bounds for the exponent of are also established for nilpotent groups G of class which extend to all similar bounds found by Moravec, P. (2008). The exponents of nonabelian tensor products of groups. J. Pure Appl. Algebra. 212(7):1840–1848.

On complete decompositions of dihedral groups

Let G be a finite non-abelian group and B1, …, Bt be nonempty subsets of G for integer t ≥ 2. Suppose that B1, …, Bt are pairwise disjoint, then (B1, …, Bt) is called a complete decomposition of G of order t if the subset product Bi1 … Bit = {bi1 … bit | bij ∈ Bij, j = 1,2, …,t} coincides with G, where {Bi1 … Bit} = {B1, …, Bt} and the Bij are all distinct. Let D2n = ‹r,s| rn = s2 = 1, rs =srn-1› be the dihedral group of order 2n for integer n ≥3. In this paper, we shall give the constructions of the complete decompositions of D2n of order t, where 2 ≤ t ≤ n.

On nil clean group rings

A ring is nil clean if each of its elements is the sum of an idempotent and a nilpotent. In Sahinkaya et al. [13], it was shown that, for a ring R and a symmetric group S 3, the group ring RS 3 is nil clean iff R and are nil clean. Let be the dihedral group of order 2n and be the generalized quaternion group of order 2n. In this paper, we investigate a more general question and completely characterize when group rings and are nil clean. It is proved that is nil clean iff, either and R is nil clean, or and RS 3 is nil clean, and a similar result is obtained for Furthermore, nil clean group rings with involution * are also investigated.

Sextic reciprocal monogenic dihedral polynomials

Let $$D_n$$ denote the dihedral group of order 2n. We find infinite $$D_6$$ -families and an infinite $$D_3$$ -family of monic irreducible reciprocal sixth-degree polynomials $$f(x)\in \mathbb {Z}[x]$$ , such that $$\{1,\theta ,\theta ^2,\theta ^3,\theta ^4,\theta ^5\}$$ is a basis for the ring of integers of $$L=\mathbb {Q}(\theta )$$ , where $$f(\theta )=0$$ .

Codes over dihedral groups

In this paper we consider the group algebra FG, where characterstic of the field F does not divide order the group G, then FG is semisimple, and hence decomposes into a direct sum of minimal ideals generated by the idempotents .We give the explicit expressions for the idempotents in the group algebra of dihedral group of order 2n for every n. We describe (2n, 2n-1, 2) MDS and (2n, 2n-2, 2) group codes for every n corresponding to the linear idempotents and in case of non linear idempotents Dihedral group codes of length 16, 20,24 are constructed.

Idempotents in the group algebra of dihedral groups

In this paper we give the explicit expressions for the idempotent elements in the group algebra of dihedral group of order 2n, for every n.

Enumerating the Walecki-Type Hamiltonian Cycle Systems

Let Kv be the complete graph on v vertices. A Hamiltonian cycle system of odd order v (briefly HCS(v)) is a set of Hamiltonian cycles of Kv whose edges partition the edge set of Kv. By means of a slight modification of the famous HCS(4n+1) of Walecki, we obtain 2n pairwise distinct HCS(4n+1) and we enumerate them up to isomorphism proving that this is equivalent to count the number of binary bracelets of length n, i.e. the orbits of Dn, the dihedral group of order 2n, acting on binary n-tuples.

Units in modular group algebra

ABSTRACTLet 𝒰(FG) denotes the unit group of FG. In this article, we compute theorder of in terms of the order of 𝒰(FG) for an arbitrary finite group G, where is the cyclic group of order 2n and F is a finite field of characteristic 2. Further, if A is an elementary abelian 2-group, then we obtain structures of 𝒰(F(G×A)) and its unitary subgroup 𝒰∗(F(G×A)), where ∗ is the canonical involution of the group algebra F(G×A). Finally, we provide a set of generators of and 𝒰(FD4m).

The unit group of some special semi-simple group algebras

Let G be a non-abelian group of order 2n which has a cyclic subgroup of index 2 and let U ( [G]) be the unit group of the group algebra [G] of G over the field containing q = pkelements. In this paper, recurrence relations are established to determine the structure of the unit group of [G] when p > 2.

The unit group of finite group algebra of a generalized dihedral group

Let [Formula: see text] be a generalized dihedral group of order 2n and 𝔽qbe a finite field having q elements. In this note, we establish the structure of the unit group of [Formula: see text] for any odd n ≥ 3. This extends a result due to Kaur and Khan [Units in 𝔽2D2p, J. Algebra Appl. 13(2) (2014) 9pp., doi: 10.1142/S0219498813500904] as well as a result due to the authors [Units in 𝔽2kD2n, Int. J. Group Theory 3(3) (2014) 25–34].

On local near-rings with Miller–Moreno multiplicative group

A near-ring R with identity is local if the set L of all its noninvertible elements is a subgroup of the additive group R +. We study local near-rings of order 2 n whose multiplicative group R * is a Miller–Moreno group, i.e., a non-abelian group all proper subgroups of which are abelian. In particular, it is proved that if L is a subgroup of index 2 m in R +, then either m is a prime number for which 2 m − 1 is a Mersenne prime or m = 1. In the first case, n = 2m, the subgroup L is elementary abelian, the exponent of R + does not exceed 4; and R * is of order 2 m (2 m − 1)). In the second case, either n < 7 or the subgroup L is abelian and R * is a nonmetacyclic group of order 2 n−1 whose exponent does not exceed 2 n−4.

Noether’s problem for the groups with a cyclic subgroup of index 4

Let G be a finite group and k be a field. Let G act on the rational function field k(xg : g ∈ G) by k-automorphisms defined by g ∙ xh = xgh for any g, h ∈ G. Noether's problem asks whether the fixed field k(G) = k(xg : g ∈ G)G is rational (i.e., purely transcendental) over k. Theorem 1. If G is a group of order 2n (n ≥ 4) and of exponent 2e such that (i) e ≥ n − 2 and (ii) \( {\zeta_{{{2^{{e - 1}}}}}} \in k \), then k(G) is k-rational. Theorem 2. Let G be a group of order 4n where n is any positive integer (it is unnecessary to assume that n is a power of 2). Assume that (i) char k ≠ 2, \( {\zeta_n} \in k \), and (ii) G contains an element of order n. Then k(G) is rational over k, except for the case n = 2 m and G ≃ Cm ⋊ C8 where m is an odd integer and the center of G is of even order (note that Cm is normal in Cm ⋊ C8); for the exceptional case, k(G) is rational over k if and only if at least one of −1; 2;−2 belongs to (k×)2.

ON THE STRUCTURE OF SOLUBLE GRADED LIE ALGEBRAS

Let A be the elementary group of order 2n and L an A-graded Lie algebra with L0 = 0. Assume that L is soluble with derived length k. It is proved that L has a series of ideals of length n all of whose quotients are nilpotent of {k, n}-bounded class.

Group elements of some semisimple finite-dimensional Hopf algebras

V. A. Artamonov and I. A. Chubarov proved a criterion under which an element of some semisimple finite-dimensional Hopf algebra is group-like. The studied Hopf algebra has only one nonone- dimensional irreducible representation. Let n be a dimension of this representation. It is shown in this paper that for odd prime n the set of group-like elements of these algebras is a cyclic group of order 2n.

Cubic one-regular graphs of order twice a square-free integer

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. In this paper, we show that a cubic one-regular graph of order 2n exists if and only if n = 3tp1p2···ps ⩾ 13, where t ⩽ 1, s ⩾ 1 and pi’s are distinct primes such that 3| (pi − 1). For such an integer n, there are 2s−1 non-isomorphic cubic one-regular graphs of order 2n, which are all Cayley graphs on the dihedral group of order 2n. As a result, no cubic one-regular graphs of order 4 times an odd square-free integer exist.

Free Pairs of Bicyclic Units in the Integral Group Ring of the Dihedral Group

We prove that if D 2n is the dihedral group of order 2n and u and v are two noncommuting bicyclic units in ℤD 2n of the same type, then the pair {u, v} always generates a free group of rank 2 except possibly if 12 divides n. The problematic cases are explicitly described. Further, it is shown that a problematic pair {u, v} generates a free group if and only if is a free point and that {u, v 2} always generates a free group. Thus we answer in the affirmative a conjecture by Jespers et al. (2002).

Some Properties of Hadamard Matrices Coming from Dihedral Groups

In [4], one of the authors introduced a method to construct Hadamard matrices of degree 8n+4 from the dihedral group of order 2n. Here we study some properties of this construction.

2-groups with few conjugacy classes

AbstractAn old question of Brauer that asks how fast numbers of conjugacy classes grow is investigated by considering the least number cn of conjugacy classes in a group of order 2n. The numbers cn are computed for n ≤ 14 and a lower bound is given for c15. It is observed that cn grows very slowly except for occasional large jumps corresponding to an increase in coclass of the minimal groups Gn. Restricting to groups that are 2-generated or have coclass at most 3 allows us to extend these computations.

Unitary units in modular group algebras of 2-groups

We study the unitary subgroupV *(F2 G) in the group algebras F2 Gof 2-groups of maximal class over the field F2of two elements. We show that there does not exist a normal complement to Gin V *(F2 G) if and only if Gis the dihedral group of order 2n(n≥ 5) or the semidihedral group of order 2n(n≥5). We also describe V *(F2 G) when the subgroup Ghas order 8 and 16 and in this case there exist a normal complement of Gin V *(F2 G).