Fast recognition of alternating groups of unknown degree

We present a constructive recognition algorithm to decide whether a given black-box group is isomorphic to an alternating or a symmetric group without prior knowledge of the degree. This eliminates the major gap in known algorithms, as they require the degree as additional input. Our methods are probabilistic and rely on results about proportions of elements with certain properties in alternating and symmetric groups. These results are of independent interest; for instance, we establish a lower bound for the proportion of involutions with small support.