S. A. Linton, G. Pfeiffer, E. F. Robertson, and N. Ruskuc,
# Groups and Actions in Finite Transformation Semigroups.

*Math. Z.* **228** (1998), 435-450.

## Abstract.

Let $S$ be a transformation semigroup of degree $n$.
To each element $s$ in $S$ we associate a permutation group
$G$_{R}(s)
acting on the image of $s$, and we find a natural
generating set for this group. It turns out that the $R$-class
of $s$ is a disjoint union of certain sets, each having
size equal to the size of $G$_{R}(s).
As a consequence, we show that two $R$-classes containing elements
with equal images have the same size, even if they do not
belong to the same $D$-class.
By a certain duality process we associate to $s$
another permutation group $G$_{L}(s) on the image of $s$,
and prove analogous results for the $L$-class of $s$.
Finally we prove that the Schützenberger group of
the $H$-class of $s$ is isomorphic to the intersection
of $G$_{R}(s) and $G$_{L}(s). The results of this paper
can also be applied in new algorithms
for investigating transformation semigroups, which
will be described in a forthcoming paper.
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