Conjugation of injections by permutations
arXiv:1003.2027 · doi:10.1007/s00233-010-9224-3
Abstract
Let X be a countably infinite set, and let f, g, and h be any three injective self-maps of X, each having at least one infinite cycle. (For instance, this holds if f, g, and h are not bijections.) We show that there are permutations a and b of X such that h=afa^{-1}bgb^{-1} if and only if |X\Xf|+|X\Xg|=|X\Xh|. We also prove a version of this statement that holds for infinite sets X that are not necessarily countable. This generalizes results of Droste and Ore about permutations.
27 pages, 4 figures