The classification of partition homogeneous groups with applications to semigroup theory

Let λ=(λ1,λ2,…) be a partition of n, a sequence of positive integers in non-increasing order with sum n. Let Ω:={1,…,n}. An ordered partition P=(A1,A2,…) of Ω has type λ if |Ai|=λi.Following Martin and Sagan, we say that G is λ-transitive if, for any two ordered partitions P=(A1,A2,…) and Q=(B1,B2,…) of Ω of type λ, there exists g∈G with Aig=Bi for all i. A group G is said to be λ-homogeneous if, given two ordered partitions P and Q as above, inducing the sets P′={A1,A2,…} and Q′={B1,B2,…}, there exists g∈G such that P′g=Q′. Clearly a λ-transitive group is λ-homogeneous.The first goal of this paper is to classify the λ-homogeneous groups (Theorems 1.1 and 1.2). The second goal is to apply this classification to a problem in semigroup theory.Let Tn and Sn denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H⩽Sn. Given a non-invertible transformation a∈Tn∖Sn and a group G⩽Sn, we say that (a,G) is an H-pair if the semigroups generated by {a}∪H and {a}∪G contain the same non-units, that is, 〈a,G〉∖G=〈a,H〉∖H. Using the classification of the λ-homogeneous groups we classify all the Sn-pairs (Theorem 1.8). For a multitude of transformation semigroups this theorem immediately implies a description of their automorphisms, congruences, generators and other relevant properties (Theorem 8.5).This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas.The paper finishes with a number of open problems on permutation and linear groups.