  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 of
this paper is to prove some general results on permutation groups and then
apply them to transformation semigroups. On groups we find the minimum number
of permutations needed to generate $k$-homogeneous groups (for $k\ge 1$); in
particular we show that $2$-homogeneous groups are $2$-generated. We also
describe the orbits of $k$-homogenous groups on partitions with $n-k$ parts,
classify the $3$-homogeneous groups $G$ whose orbits on $(n-3)$-partitions are
invariant under the normalizer of $G$ in $S_n$, and describe the normalizers of
$2$-homogeneous groups in the symmetric group. Then these results are applied
to extract information about transformation semigroups with given group of
units, namely to prove results on their automorphisms and on the minimum number
of generators. The paper finishes with some problems on permutation groups,
transformation semigroups and computational algebra.
