Article,
A note on automorphisms of coset digraphs.
Result. Math., 41 (1-2): 26-33 (2002)
Abstract
Let $G$ be a strongly connected digraph whose set of arcs admits a
decomposition into oriented cycles. Let $P$ be a partition of the
cycles of such a decomposition, where the cycles in a class of $P$ are
vertex disjoint. The paper investigates the structure of $\cal
P$-automorphism groups of $G$, which preserve the partition $P$. \par
Each class of $P$ can be identified with the permutation on $V(G)$ with
the corresponding cycle decomposition. Then $G$ can be represented as a
Schreier coset digraph of the permutation group $T$ generated by these
permutations. The group of automorphisms which leave each class of $P$
invariant is isomorphic to the factor group $N_T(T_v)/T_v$, where $T_v$ is
the stabilizer of a vertex and $N_T(T_v)$ denotes the normalizer of $T_v$ in
$T$. The structure of a larger group of automorphisms leaving $P$
invariant is also given as a subgroup of $(Aut_\Delta(T)\times_s
T)/T_v$, where $Aut_\Delta$ is a group of automorphisms of $T$
leaving the generating set of $T$ invariant and $\times_s$ denotes the
semidirect product. This larger group is the full $P$-automorphism
group when $N_T(T_v)/T_v$ acts transitively on $V(G)$.These results
extend previous work of the author et al. for undirected graphs Zbl
0722.05036, Zbl 0722.05037. Similar results for regular digraphs where
obtained in O. Serra and M. A. Fiol Research and Lecture Notes
in Mathematics. 413-420 (1991; Zbl 0945.05517).
