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 |A_i|=λ_i.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 A_ig=B_i 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 T_n and S_n denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H<=S_n. Given a non-invertible transformation a in T_n-S_n and a group G<=S_n, 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 S_n-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.