Fast recognition of alternating groups of unknown degree

TL;DR

This paper introduces a probabilistic algorithm for recognizing whether a black-box group is an alternating or symmetric group without knowing its degree beforehand, filling a key gap in existing methods.

Contribution

The authors develop a constructive recognition algorithm that operates without prior degree information, advancing the computational group theory field.

Findings

Established a lower bound for the proportion of involutions with small support in these groups.

Provided a probabilistic method that is effective for groups of unknown degree.

Filled a major gap in recognition algorithms by removing the need for degree input.

Abstract

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.