  The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may
exhibit symmetries in its form that allow for reduction of order through
substitution or a change of variables. Such form symmetries can be defined
generally using the semiconjugate relation on a group which yields a reduction
of order through the semiconjugate factorization of the difference equation of
order $k+1$ into equations of lesser orders. Different classes of equations are
considered including separable equations and homogeneous equations of degree 1.
Applications include giving a complete factorization of the linear
non-homogeneous difference equation of order $k+1$ into a system of $k+1$ first
order linear non-homogeneous equations in which the coefficients are the
eigenvalues of the higher order equation. Form symmetries are also used to
explain the complicated multistable behavior of a separable, second order
exponential equation.

Dynamical Systems;Exactly Solvable and Integrable Systems