2013/04/27 by Jorge André, João Araújo, André, Jorge +3
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR
paper · pdf · doi:10.48550/arxiv.1304.7391
arxiv created 2013/04/27 · arxiv updated 2013/04/30
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. The second goal is to apply this classification to a problem in semigroup theory. Let \trans and \sym denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H≤ \sym. Given a non-invertible transformation a∈ \trans∖ \sym and a group G≤ \sym, 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 \sym-pairs. 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.