ACTA issues

Permutation groups, pattern involvement, and Galois connections

Erkko Lehtonen, Reinhard Pöschel

Acta Sci. Math. (Szeged) 83:3-4(2017), 355-375

Abstract. There is a connection between permutation groups and permutation patterns: for any subgroup $G$ of the symmetric group $\symm{\ell }$ and for any $n \geq\ell $, the set of $n$-permutations involving only members of $G$ as $\ell $-patterns is a subgroup of $\symm{n}$. Making use of the monotone Galois connection induced by the pattern avoidance relation, we characterize the permutation groups that arise via pattern avoidance as automorphism groups of relations of a certain special form. We also investigate a related monotone Galois connection for permutation groups and describe its closed sets and kernels as automorphism groups of relations.

DOI: 10.14232/actasm-017-510-4

AMS Subject Classification (1991): 08A40, 05A05

Keyword(s): permutation patterns, Galois connections

Received February 1, 2017. (Registered under 10/2017.)