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