The divided differences in theory of differential-difference equatians and in theory of groups

Complete description of nonlinear groups of all automorphisms of finitely generated relatively free groups with a nontrivial law is gave. It is shown that finitely generated soluble-by-finite subgroups of such groups is cause of nonlinearity of these groups. A wide class of such subgroups for any finite rank of the relatively free group is gave. Method of decomposition of the polynomial with integral coefficients into simple (in $\mathbb{Q}[x]$) factors, to prove the last result, is proposed. It is based on property of divided differences. The proof of last result of the article is based on another property of divided differences. It is proved that for a complex matrix A (of degree $n$) with real spectrum the matrix exponential $\exp(A)$ does not belong to linear span of matrices $A^0,A^1,\dots,A^{n-2}$.