Primitive Permutation Groups and Strongly Factorizable Transformation Semigroups

2019/10/18 by Araújo, João, Bentz, Wolfram, Cameron, Peter J.
#20B15 20M20 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1910.08335

Abstract

Let Ω be a finite set and T(Ω) be the full transformation monoid on Ω. The rank of a transformation t∈ T(Ω) is the natural number |Ωt|. Given A⊆ T(Ω), denote by ⟨ A⟩ the semigroup generated by A. Let k be a fixed natural number such that 2≤ k≤ |Ω|. In the first part of this paper we (almost) classify the permutation groups G on Ω such that for all rank k transformation t∈ T(Ω), every element in S_t:=⟨ G,t⟩ can be written as a product eg, where e²=e∈ S_t and g∈ G. In the second part we prove, among other results, that if S≤ T(Ω) and G is the normalizer of S in the symmetric group on Ω, then the semigroup SG is regular if and only if S is regular. (Recall that a semigroup S is regular if for all s∈ S there exists s'∈ S such that s=ss's.) The paper ends with a list of problems.

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