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Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is$k$-homogeneous if it has only one orbit on the sets of size $k$. The aim ofthis paper is to prove some general results on permutation groups and thenapply them to transformation semigroups. On groups we find the minimum numberof permutations needed to generate $k$-homogeneous groups (for $k\ge 1$); inparticular we show that $2$-homogeneous groups are $2$-generated. We alsodescribe the orbits of $k$-homogenous groups on partitions with $n-k$ parts,classify the $3$-homogeneous groups $G$ whose orbits on $(n-3)$-partitions areinvariant under the normalizer of $G$ in $S_n$, and describe the normalizers of$2$-homogeneous groups in the symmetric group. Then these results are appliedto extract information about transformation semigroups with given group ofunits, namely to prove results on their automorphisms and on the minimum numberof generators. The paper finishes with some problems on permutation groups,transformation semigroups and computational algebra.

Orbits of Primitive $k$-Homogenous Groups on $(n-k)$-Partitions with Applications to Semigroups