Let p be a prime. We denote by Sn the symmetric group of degree n, by An the alternating group of degree n and by Fp the field with p elements. An important concept of modular representation theory of a finite group G is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring FqG, where q is a prime...

Recent work on central units of integral group rings is surveyed. In particular we present two methods of constructing central units, induction and lifting, and demonstrate how these constructions can often be used to find generators for large subgroups in the full group of central units of an integral group ring.

In this paper a finite set of generators is given for a subgroup of finite index in the group of central units of the integral group ring of a finitely generated nilpotent group. In this paper we construct explicitly a finite set of generators for a subgroup of finite index in the centre Z(U(ZG)) of the unit group U(ZG) of the integral group ring ZG of a finitely generated nilpotent group G. Ri...

In the title bis-tolane derivative, C(29)H(24)O(2), the central benzene ring forms dihedral angles of 29.12 (9) and 26.46 (9)° with the other two benzene rings. The dihedral angle between two terminal benzene rings is 55.58 (8)°.

In the title mol-ecule, C13H7N3O8, the phenyl and benzene rings are rotated from the mean plane of the central ester group by 18.41 (9) and 81.80 (5)°, respectively. The dihedral angle between the rings is 80.12 (14)°. In the crystal, mol-ecules are linked by weak C-H⋯O inter-actions, forming helical chains along [010].

In the title compound, C(17)H(10)Cl(2)N(2)O(3)S, the thio-phene ring and the central benzene ring are almost coplanar [dihedral angle = 8.44 (3)°], while the dihedral angle between the two benzene rings rings is 77.49 (9)°. The crystal packing is stabilized by inter-molecular C-H⋯O hydrogen bonds.

In the title compound, C(17)H(14)N(2)O(2), the central pyridazine ring forms dihedral angles of 47.29 (5) and 88.54 (5)° with the benzene rings, while the dihedral angle between the benzene rings is 62.68 (6)°. In the crystal, molecules are linked by two weak C-H⋯O hydrogen bonds and three weak C-H⋯π inter-actions.

Individual mol-ecules of the title compound, C(24)H(19)N, do not participate in hydrogen-bonding inter-actions due to the steric bulk of the phenyl rings ortho to the amine. The dihedral angles between the central ring and the pendant rings are 68.26 (10), 55.28 (10) and 30.61 (11)°.