  We discuss a general method by which a higher order difference equation on a
group is transformed into an equivalent triangular system of two difference
equations of lower orders. This breakdown into lower order equations is based
on the existence of a semiconjugate relation between the unfolding map of the
difference equation and a lower dimensional mapping that unfolds a lower order
difference equation. Substantial classes of difference equations are shown to
possess this property and for these types of equations reductions of order are
obtained. In some cases a complete semiconjugate factorization into a
triangular system of first order equations is possible.
