Let P be the derived subgroup of (G( MM _k)).
It remains to show that its kernel is the derived subgroup of (mathcal {K}_p).
In a nilpotent group, the derived subgroup is contained in the Frattini subgroup [2, 5.2.16], so (G'leqPhi(G)) and hence (G/Phi(G)) is cyclic.
It was concluded that individuals with high-functioning autism have a cognitive profile that resembles that of an empirically derived subgroup of schizophrenia patients but that does not resemble profiles found in other schizophrenia subgroups.
Note also that although our groups are not linear, they are (Abelian of prime exponent -by-linear sincexponent -by-linear of class 3 with thexponent -by-linearbgroup Abelian of prime exponent (one can asince they the exponent is equareto any prime number, solvable
It is proved in [27, Theorem 6] (see also Equation (5.4) in [28]) that the derived subgroup ([mathcal {K}_p, mathcal {K}_p]) of (mathcal {K}_p) is the set of elements defined by tableaux in which (a_0) and the coefficient (c_{p-1, p-1, ldots, p-1}) at the eldest term (x_1^{p-1}x_2^{p-1}cdots x_n^{p-1}) are equal to zero for every n.
Moreover, the five extraintestinal pathogenic E. coli strains clustered together, and within the phylogroup B2, they were the most derived subgroup.
This also holds true for sea urchins (Echinodermata: Echinoidea), in which a highly specialized gastric caecum can be found in members of a derived subgroup, the Irregularia (cake urchins, sea biscuits, sand dollars, heart urchins, and related forms).
This cross validation study examined the reliability of the Multidimensional Pain Inventory (MPI) and the stability of the Multidimensional Pain Inventory Classification System of the empirically derived subgroup classification obtained by cluster analysis in chronic musculoskeletal pain.
Two studies examined the cluster stability of the empirically derived subgroup classification based on the Multidimensional Pain Inventory and found that retest resulted in 65 to 70% of patients being assigned to the same cluster [ 29, 30].
