# London Algebra Colloquium abstract

## Partition-homogeneity and applications

### Professor Peter J. Cameron (Queen Mary), 14th February 2013

#### Abstract

(Joint work with João Araújo.) Martin and Sagan defined a permutation group of degree *n*
to be λ-transitive, where λ is a partition of *n*, if it acts transitively
on ordered set-partitions of shape λ. There is an obvious weakening to transitivity on
unordered partitions, which by analogy with classical concepts we call λ-homogeneity.
This arises in a problem on transformation semigroups. I will prove a partition analogue of
the Livingstone–Wagner Theorem by finding all the groups which are λ-homogeneous
but not λ-transitive, and explain how this is applied in semigroup theory.