## Abstract

The purpose of this paper is to advance our knowledge of two of the most classic and popular topics in transformation semigroups: automorphisms and the size of minimal generating sets. In order to do this, we examine the

*k*-homogeneous permutation groups (those which act transitively on the subsets of size_{k}of their domain*X*) where |*X*|=*n*and k<*n*/2. In the process we obtain, for*k*-homogeneous groups, results on the minimum numbers of generators, the numbers of orbits on*k*-partitions, and their normalizers in the symmetric group. As a sample result, we show that every finite 2-homogeneous group is 2-generated. Underlying our investigations on automorphisms of transformation semigroups is the following conjecture:*If a transformation semigroup S contains singular maps, and its group of units is a primitive group G of permutations, then its automorphisms are all induced (under conjugation) by the elements in the normalizer of G in the symmetric group.**For the special case that*

*S*contains all constant maps, this conjecture was proved correct, more than 40 years ago. In this paper, we prove that the conjecture also holds for the case of semigroups containing a map of rank 3 or less. The effort in establishing this result suggests that further improvements might be a great challenge. This problem and several additional ones on permutation groups, transformation semigroups and computational algebra, are proposed in the end of the paper.Original language | English |
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Pages (from-to) | 105-136 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 1 |

Early online date | 3 May 2018 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

## Keywords

- Permutation group
- Transformation semigroup
- Automorphism

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