A , nite group G, in which two randomly chosen subgroups H and K commute, has been classi, ed by Iwasawa in 1941. It is possible to de, ne a probabilistic notion, which \measures the distance" of G from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree gsd(G) for two arbitrary Sublattices S(G) and T(G) of the lattice of subgroups L(G) of G. Upper and lower bounds for gsd(G) are shown and we study the behaviour of gsd(G) with respect to subgroups and quotients, showing new numerical restrictions.

In this research the magnetoelastic properties of Nd6Fe13-xCoxCu (x=0 and 1) compound are investigated. Analysis of X-ray diffraction patterns indicates that a single phase is formed approximately for x=0 and the x=1 sample is multi-phase. The lattice parameters were decreased, the Néel temperature of main phase and the Curie temperature of impurity phase was increased with Co content. Due to the presence of Nd2Fe17-yCoy ferromagnetic phase in the sample with x=1 the change of the anisotropy and increase of exchange effects was observed in the magnetic and magnetoelastic measurements. Thermal expansion, longitudinal (ll) and transverse (lt) magnetostriction were measured using the strain gauge method in the selected temperatures range of 80- 500 K under applied magnetic fields up to 1.5 T. Anomaly and Invar effect are observed in the linear thermal expansion and a (T) curves at the Néel temperature. The linear spontaneous magnetostriction decreases sharply by approaching the Néel temperature and also shows the short-range magnetic ordering effects when antiferromagnetic to paramagnetic transition occurs. In the low field region, the absolute values of anisotropic magnetostriction of Nd6Fe13Cu compound are small and then start to increase with applied magnetic field. Each isofield curve of the anisotropic magnetostriction passes through a minimum and then approaches to zero with increasing temperature. This magnetostriction compensation arises from the difference in the magnetoelastic coupling constants of the Sublattices in this compound.