The lower central and derived series of the braid groups of the projective plane

In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups B(n)(RP(2)) of the projective plane RP(2) were originally studied by Van Buskirk during the 1960s. and are of particular interest due to the fact that they have torsion. The group B(1)(RP(2)) (resp. B(2)(RP(2))) is isomorphic to the cyclic group Z(2) of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If n > 2, we first prove that the lower central series of B(n)(RP(2)) is constant from the commutator subgroup onwards. We observe that Gamma(2)(B(3)(RP(2))) is isomorphic to (F(3) X Q(8)) X Z(3), where F(k) denotes the free group of rank k, and Q(8) denotes the quaternion group of order 8, and that Gamma(2)(B(4)(RP(2))) is an extension of an index 2 subgroup K of P(4)(RP(2)) by Z(2) circle plus Z(2). As for the derived series of B(n)(RP(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group B(n)(RP(2)) being finite and soluble for n <= 2, the critical cases are n = 3, 4. We are able to determine completely the derived series of B(3)(RP(2)). The subgroups (B(3)(RP(2)))((1)), (B(3)(RP(2)))((2)) and (B(3)(RP(2)))((3)) are isomorphic respectively to (F(3) x Q(8)) x Z(3), F(3) X Q(8) and F(9) X Z(2), and we compute the derived series quotients of these groups. From (B(3)(RP(2)))((4)) onwards, the derived series of B(3)(RP(2)), as well as its successive derived series quotients, coincide with those of F(9). We analyse the derived series of B(4)(RP(2)) and its quotients up to (B(4)(RP(2)))((4)), and we show that (B(4)(RP(2)))((4)) is a semi-direct product of F(129) by F(17). Finally, we give a presentation of Gamma(2)(B(n)(RP(2))). (C) 2011 Elsevier Inc. All rights reserved.

Formation of out of plane oxime metallacycles in [Cu(2)] and [Cu(4)] complexes

The reaction Of Cu(ClO(4))(2)center dot 6H(2)O with dimethylglyoxime (H(2)dmg) in a 1:1 mole ratio in aqueous methanol at room temperature affords the dinuclear complex [Cu(2)(mu-Hdmg)(4)] (1). Reaction of 1 with [Cu(bpy)(H(2)O)(2)](ClO(4))(2) (bpy = 2,2`-bipyridine) in a 1:1 mole ratio in aqueous methanol at room temperature yields the tetranuclear complex [Cu(2)(mu-HdMg)(2)(mu-dMg)(2)(bpy)(2)(H(2)O)(2)](ClO(4))(2) (2). The direct reaction of Cu(ClO(4))(2)center dot 6H(2)O with H(2)dmg and bpy in a 2:21 mole ratio in aqueous methanol at room temperature also yields 2 quantitatively. The complexes 1 and 2 were structurally characterized by X-ray crystallography. Unlike the binding in Ni/Co-dmg, two different types of N-O bridging modes during the oxime based metallacycle formation and stacking of square planar units have been identified in these complexes. The neutral dinuclear complex 1 has CuN(4)O coordination spheres and complex 2 consists of a dicationic [Cu(2)(mu-HdMg)(2)(mu-dMg)(2)(bpy)(2)(H(2)O)(2)](2+) unit and two uncoordinated ClO(4)(-) anions having CuN(4)O and CuN(2)O(3) coordination spheres. The two copper(II) ions are at a distance of 3.846(8) angstrom in 1 for the trans out of plane link and at 3.419(10) and 3.684(10) angstrom in 2 for the trans out of plane and cis in plane arrangements, respectively. The average Cu-N(oxime) distances are 1.953 and 1.935 angstrom, respectively. The average basal and apical Cu-N(oxime) distances are 1.945, 2.295 and 2.429 angstrom. The UV-Vis spectra of 2 is similar to the spectrum of the reaction mixture of 1 and [Cu(bpy)(H(2)O)(2)](2+). Variable temperature magnetic properties measurement shows that the interaction between the paramagnetic copper centers in complex I is antiferromagnetic in nature. The EPR spectra of frozen solution of the complexes at 77 K consist of axially symmetric fine-structure transitions (Delta M(S) = 1) and half-field signals (Delta M(S) = 2) at ca. 1600 G, suggesting the presence of appreciable Cu-Cu interactions. (C) 2009 Elsevier Ltd. All rights reserved.