Groups Generated by Pattern Avoiding Permutations

Abstract

We study groups generated by sets of pattern avoiding permutations. In the first part of the paper, we prove some general results concerning the structure of such groups. In particular, we consider the sequence ( G n ) n ≥ 0 , where G n is the group generated by a subset of the symmetric group S n consisting of permutations that avoid a given set of patterns. We analyze under which conditions the sequence ( G n ) n ≥ 0 is eventually constant. Moreover, we find a set of patterns such that ( G n ) n ≥ 0 is eventually equal to an assigned symmetric group. Furthermore, we show that any non-trivial simple group cannot be obtained in this way and describe all the non-trivial abelian groups that arise in this way. In the second part of the paper, we carry out a case-by-case analysis of groups generated by permutations avoiding a few short patterns.