The Topology of Algebra: Combinatorics of Squaring

We study the graph each of whose edges connects an element of a given ring with the square of itself. For a finite commutative group (e.g., for the multiplicative group of coprime residue classes modulo a positive integer), we describe this graph explicitly: each of its connected components is an oriented attracting cycle equipped with identical $2^k$-vertex rooted trees of special form whose roots reside on the cycle. We also compute the graphs of permutation groups on not too many elements and of the subgroups of even permutations; the connected components of these graphs are also uniformly equipped cycles.