  In this paper two important aspects related to Caputo fractional-order
discrete variant of a class of maps defined on the complex plane, are
analytically and numerically revealed: attractors symmetry-broken induced by
the fractional-order and the sensible problem of determining the right
bifurcation diagram of discrete systems of fractional-order. It is proved that
maps of integer order with dihedral symmetry or cycle symmetry loose their
symmetry once they are transformed in fractional-order maps. Also, it is
conjectured that, contrarily to integer-order maps, determining the bifurcation
diagrams of fractional-order maps is far from being a clarified problem. Two
examples are considered: dihedral logistic map and cyclic logistic map.
