Let p be an odd prime and Λ=Z[D2p] the integral group ring of dihedral D2p order 2p. The syzygies Ωr(Z) are stable classes intermediate modules in a free Λ-resolution trivial module. We will discuss explicitly interaction under − ⊗Z−.

‎in this paper‎, ‎we obtain the dimensions of symmetry classes of polynomials associated with‎ ‎the irreducible characters of the dihedral group as a subgroup of‎ ‎the full symmetric group‎. ‎then we discuss the existence of o-basis‎ ‎of these classes‎.

Dihedral codes, particular cases of quasi-cyclic have a nice algebraic structure which allows to store them efficiently. In this paper, we investigate it and prove some lower bounds on their dimension minimum distance, in analogy with the theory BCH codes. This us construct dihedral codes prescribed distance. binary case, present examples optimal obtained by construction.

Let G = Cm p o C2 be a generalized dihedral group for an odd prime and natural number m, L M(G; 2) the RA2 loop obtained from F finite field of characteristic 2. For algebra F[L], we determine Jacobson radical J(F[L]) F[L] Wedderburn decomposition F[L]=J(F[L]). The structure 1 + is also determined.

In the title compound, C16H10O8·H2O, the dihedral angle between the benzene rings is 71.59 (8)°. The COOH groups make dihedral angles of 10.3 (2), 30.8 (2), 11.3 (2) and 42.3 (2)° with their attached rings. In the crystal, O-H⋯O hydrogen bonds link the components forming a three-dimensional supra-molecular network.

In the title mol-ecule, C(23)H(20)N(4)O(2)S, the triazole ring forms dihedral angles of 150.3 (2), 77.3 (2) and 77.6 (2)°, respectively, with the furan ring and the phenyl rings. The furan ring is almost perpendicular to the central phenyl ring, making a dihedral angle of 86.0 (3)°.