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\title {\bf Circulant homogeneous factorisations  of \\complete digraphs $\K_{p^d}$ with $p$ an odd prime}

\author{Jing Xu\thanks{This work
  was supported by NSFC (project number   11371259).}\\
\small Department of Mathematics\\[-0.8ex]
\small Capital Normal University\\[-0.8ex]
\small Beijing 100048,  China\\
\small\tt xujing@cnu.edu.cn}

\date{\dateline{Sep 21, 2016}{May 15, 2017}{Jun 2, 2017}\\
\small Mathematics Subject Classifications: 05C25, 20B25}

\begin{document}

\maketitle

\begin{abstract}Let $\F=(\K_{n},\P)$ be  a circulant   homogeneous factorisation of index $k$, that means $\P$ is a partition of the  arc set of  the complete digraph $\K_n$ into $k$ circulant factor digraphs such that there exists   $\s\in S_n$ permuting the factor circulants transitively amongst themselves. Suppose further such an element $\s$ normalises the cyclic regular automorphism group of these circulant factor digraphs, we say $\F$ is normal.  Let $\F=(\K_{p^d},\P)$ be a circulant homogeneous factorisation of index $k$ where $p^d$,  ($d\ge 1$) is an odd prime power. It is shown in this paper that either $\F$ is normal or $\F$ is a lexicographic product of two smaller circulant homogeneous factorisations.

\end{abstract}


\section{Introduction}
Let $\G=(V,A)$ be a \emph{digraph} with vertex set $V=V(\G)$ and arc set $A=A(\G)$. A \emph{factorisation} of $\G$ is a partition $\mathcal{P}=\{P_1,\ldots,P_k\}$ of $A$, and is denoted by the pair $(\G,\mathcal{P})$.  This gives rise to \emph{factor digraphs}, $\G_i=(V, P_i)$ for $i=1,\ldots,k$. Moreover the integer $k=|\mathcal P|$ is called the \emph{index} of the factorisation and $|V|$ is called the \emph{order} of the factorisation.

An \emph{automorphism} of a factorisation $(\G, \P)$ is an automorphism of the digraph $\G$ that  preserves the partition $\P$. The automorphism group $\Aut(\G,\P)$ consists of all automorphisms of $(\G, \P)$.
A   factorisation  $(\G, \mathcal{P})$  is called \emph{transitive}  if   $\Aut (\G, \mathcal{P})$  induces a transitive action on $\mathcal P$;  further $(\G, \mathcal{P})$ is called \emph{homogeneous} if it is transitive and in addition the kernel of $
\Aut(\G, \mathcal P)$ acting on $\mathcal P$ is transitive on the vertex set $V(\G)$.
 That is  there exists a subgroup $G\le \Aut(\G,\P)$ that  permutes the parts $P_i$ transitively,  and the kernel $M(=G\cap \Aut(\G_1)\cap \cdots \cap \Aut(\G_k))$ of $G$ acting on $\mathcal P$ is vertex transitive on each of the $\G_i$. Hence $G$ induces isomorphisms between each pair of the factor digraphs. To emphasise the groups $M$ and $G$, we say that the factorisation $(\G, \mathcal P)$ is \emph{$(M,G)$-homogeneous.} Such a factorisation is also denoted by $(\G,\mathcal P, M, G)$.  We say a  homogeneous factorisation $(\G,\mathcal P, M, G)$ is\emph{ cyclic} if the induced group $G^{\mathcal P}$ is cyclic.

A digraph   is called a {\it circulant}  if it has a cyclic group
of automorphisms which is regular on the vertex set.
 (A   permutation group  is  \emph{regular} if  it is
transitive and the only element that fixes a point is the identity.)
Let $(\G, \mathcal P)$ be $(M,G)$-homogeneous, suppose further that $M$ contains a regular cyclic subgroup. Then all of the factor digraphs $\G_i$ are  circulants relative to the same cyclic regular subgroup, and $(\G, \mathcal P)$ is called a \emph{circulant }$(M,G)$-homogeneous factorisation.


We  denote by $\K_n$ the complete digraph
on $n$ vertices in which each ordered pair of distinct
vertices is an arc. Homogeneous factorisations of $\K_n$ with index 2   arise from pairs $\G, \ov \G$, where $\G$ is a vertex transitive digraph isomorphic to its complement $\ov\G$, that is, $\G$ is a vertex transitive \emph{self-complementary } directed graph. Suppose further that the factors $\G, \ov \G$ are undirected. Then $\G$ is a vertex transitive self-complementary  undirected graph.
Moreover it is easy to see that self-complementary circulants $(\G, \ov \G)$ correspond  to a circulant homogeneous factorisation arising from this pair.
A better understanding of vertex transitive self-complementary undirected or directed graphs is a principal motivation for studying homogeneous factorisations of $\K_n$, see for example \cite{lipraeger1, lipraeger}.
 In \cite{lsx}, self-complementary circulants of prime power order have been classified.  Our main purpose of this paper is to
 classify circulant homogeneous factorisations  of complete digraphs with prime power order, which can be viewed as a generalization of the result of \cite{lsx}.



 Let $\F=(\K_{n},\P)$ be  a circulant   homogeneous factorisation. An automorphism $\t(\in \Aut(\F))$ is called  a \emph{ cyclic isomorphism} of  $\F$ if $\t$ is transitive on $\P$. Suppose further that $\F$ is $(M, G)$-homogeneous. It is proved in \cite[Theorem 4.1]{lipraeger} that  $\F$ must be cyclic,  that is there exists $\s\in G\setminus M$ such that $G=\la M,\s\ra$ and $G^{\P}=\la\s^{\P}\ra$.  Such an element  $\s$ is a cyclic isomorphism of $\F$, and we often refer to $(\K_n, \P, M, \s)$ as this factorisation when we wish to emphasise this cyclic isomorphism.
 Moreover, let $Z_{n}(\le M)$ be the regular subgroup on $V(\K_n)$ and  we identify the vertex set with this regular group $Z_n$.
   Suppose further that  there exists   a cyclic isomorphism  $\s$ fixing point 1 (the identity element of $Z_n$)  and normalising  the regular cyclic subgroup $Z_{n}$. We say such a circulant  homogeneous factorisation is  \emph{normal}. In this case, $\s$ can be viewed as an
automorphism of the cyclic group $Z_{n}$, and we can construct such  factorisations easily, see Construction \ref{con1} for more details. We also give   examples of non-normal circulant homogeneous factorisations   in Proposition \ref{nonnormal}.


We next   present a lexicographic product construction of   two circulant homogeneous factorisations which is analogous to the lexicographic product construction of two digraphs. For more information see \cite{GLPP, cons-sc}.

 For two digraphs $\G_i=(V_i,A_i)$ with $i=1, 2$,   we denote by $A_1[A_2]$ the set of all pairs $((u_1, u_2), (v_1, v_2))$
 such that either $(u_1, v_1)\in A_1$, or $u_1=v_1$ and $(u_2, v_2)\in A_2$. Then the \emph{lexicographic product } $\G_1[\G_2]$ is defined as the digraph with vertex set $V_1\times V_2$ and arc set $A_1[A_2]$.

For $i=1,2$, let $\mathcal F_i=(\K_{n_i}, \P_i,  M_i, \s_i)$  be a circulant homogeneous factorisation of index $k$   where $\s_i$ is a  cyclic isomorphism of $\F_i$. Let $\P_i=\{P_{i,1},\ldots,P_{i,k}\}$  and suppose that $\s_i: P_{i,j}\mapsto P_{i,j+1}$
(reading the second subscript modulo $k$). Write the vertex set $\K_{n_1n_2}$ as $V(\K_{n_1})\times V(\K_{n_2})$ and let $
 \F_1[\F_2] =(\K_{n_1 n_2},\P)$ where   $\P=\{P_{1,j}[P_{2,j}]| 1\le j\le k\}$.  It is  proved in Lemma \ref{lexi} that $\F_1[\F_2]$ is a circulant homogeneous factorisation of index $k$ and $\F_1[\F_2]$ is called the \emph{lexicographic product} of   $\mathcal F_1$ and $\F_2$. Let $\G_{i,j} (i=1,2, j=1,\ldots,k)$ be the factors of $\F_i$. Then $
 \G_{1,j}[\G_{2,j}]$ are factors of $\F_1[\F_2]$.


In this paper, we will classify circulant homogeneous  factorisations of a complete digraph $\K_{p^d}$ where $p$ is an odd prime and $d\ge 1$. The main theorem is the following.
\begin{theorem}\label{mainthm}
Let $\F=(\K_{p^d},\P)$ be a circulant homogeneous factorisation of index $k$ where $p^d (d\ge 1)$ is an odd prime power. Then either $\F$ is normal or $\F=\F_1[\F_2]$ is a lexicographic product where $\F_i  $ is a circulant homogeneous factorisation  of  index $k$  and order $p^{d_i}$ for $i=1,2$ and $d=d_1+d_2$ ($d_1,d_2\ge 1$).
\end{theorem}




\section{Preliminary results}
From now on, we always assume that $p^d (d\ge 1)$  is an odd prime
power.

 A finite permutation group is called a
\emph{c-group} if it contains a cyclic regular subgroup.   A precise list of primitive c-groups  is given in the following lemma.


\begin{lemma}\label{primitive-c-gps}{\rm(\cite[Theorem 1.2]{lipraeger})}
Suppose that $X$ is a primitive permutation group on $\O $ and $X$
contains a cyclic regular subgroup. Then one of the following holds:
\begin{itemize}
\item[(i)] $|\O |=q$, and $X\le\AGL(1,q)$, where $q$ is a prime;

\item[(ii)] $|\O |=4$, and $X= Sym(\O)$;

\item[(iii)] $X$ is almost simple and $2$-transitive on $\O $.
\end{itemize}
\end{lemma}


\begin{corollary}\label{X-prim}
Suppose that $X$ is a solvable primitive c-group on $\O$
where $|\O|=p^d$ is an odd prime power. Then $|\O|=p$ and $\Z_p\le X\le\AGL(1,p)$.
\end{corollary}



Let $Z_n$ be a cyclic group of order $n$, considered in its action (by multiplication) as a subgroup of the symmetric group $\Sym (Z_n)$.
A \emph{$Z_n$-circulant} is a Cayley digraph $\G=\Cay(Z_n,S)$
with vertex set $Z_n$ and arc set $A(\G)=\{(g,sg)\mid
 g\in Z_n, s\in S\}$, for some non-empty subset $S$ of $Z_n\setminus \{1\}$. We also denote by $\hat Z_{n}$  the  right regular representation of the group $Z_{n}$.
Then each $Z_n$-circulant $\G$ admits $\hat Z_n$ as
a subgroup of automorphisms. Consider also $\Aut(Z_n)$ as a subgroup of $\Sym(Z_n)$ in its natural action.
Then $\Aut(Z_n)$ normalises $\hat Z_n$ in $\Sym(Z_n)$, and
$\Aut(Z_n)\cap \Aut(\G)$ is equal to $\Aut(Z_n,S):=
\{\s\in\Aut(Z_n)\mid S^{\s}=S\}$. In fact  the normaliser $N_{\Aut(\G)}(\hat Z_n)=\hat Z_n\rtimes\Aut(Z_n,S)$,
see for example \cite{godsil,Xu-survey}. The Cayley digraph $\G=\Cay(Z_n, S)$ is said to be a
\emph{normal circulant} if $\hat Z_n$ is normal in $\Aut(\G)$, or equivalently, if
$\Aut(\G)=\hat Z_n\rtimes \Aut(Z_n,S)$.



A circulant is called \emph{arc-transitive} if its automorphism group is transitive on the arc set. The finite arc-transitive circulants were classified independently  by Istv\'an
Kov\'acs~\cite{kovacs} and Cai Heng Li~\cite{li-arc-circ} in 2004.
The following result concerning   arc-transitive circulants of order
$p^d$ is an immediate corollary of Theorem 1.3 in \cite{li-arc-circ}, (just note that in \cite[Theorem 1.3]{li-arc-circ}, the orders of the deleted lexicographic product type digraphs cannot  be a prime power).


\begin{theorem}\label{arc-circ}
Let $\G=\Cay(Z_{p^d}, S)$ be a connected arc transitive  directed circulant  of
order $p^d$ where $p$ is an odd prime and $d\ge 1$ is an integer.
Then one of the following holds:
\begin{enumerate}
\item[(i)] $\G$ is a complete digraph.

\item[(ii)] $\G$ is a normal circulant.

\item[(iii)] There exists an arc-transitive  circulant $\si$ of order
$p^{d-i}$ such that $\G=\si[\ov K_{p^i}]$ where $1\le i<d$. Let
$Z_{p^i}\le Z_{p^d}$ be the subgroup of order $p^i$. Then
$sZ_{p^i}\subseteq S$ for any $s\in S$.
\end{enumerate}
\end{theorem}
Let $\O$ be the vertex set of $\K_n$. Then a factorisation $(\K_n, \P)$
is also simply denoted by $(\O, \P)$.

Let $\F=(\K_{n}, \P)$ be   a $(M, G)$-homogeneous factorisation  of index $k$  where $\P=\{P_1,\ldots, P_k\}$, and assume that $G$ is imprimitive on the vertex set of $\K_n$. Let $B$  be a block of $G$. Let $P_i^B=P_i\cap (B\times B)$ and $\P_{B}=\{P_1^B, P_2^B,\ldots,P_k^B\}$. Then the factorisation $(B,\P_B)$ is called  \emph{the induced sub-factorisation} of $\F$ on the block $B$.





 \begin{lemma} \label{lpblock} {\rm(\cite[Lemma 4.1]{lipraeger1})}
 Let $(\K_{n}, \P, M, G)$  be a   homogeneous factorisation  of index $k$  and let $B$  be a nontrivial block of   $G$. Then the induced sub-factorisation $(B,\P_B)$ is an $(M_B^B, G_B^B)$-homogeneous factorisation of index $k$. Further, $G^{
 \P}$ is permutationally isomorphic to $G_B^{\P_B}$.
\end{lemma}




Let $(\K_n, \mathcal P)$ be a circulant $(M,G)$-homogeneous factorisation of index $k$ where $\P=\{P_1,\ldots, P_k\}$ and   identify
 the vertex set with the regular group $Z_n$.
 Choose the point $1\in Z_n$, let $P_i(1)=\{\a\in  Z_n| (1,\a)\in P_i\}$ and let $\P(1)=\{P_1(1),\ldots,P_k(1)\}$. Then $\P(1)$ is a partition of $Z_n\setminus \{1\}$, and the factor digraphs $\G_i=\Cay(Z_n, P_i(1))$. The   lemma below gives a relation between the partition $\P$ and $\P(1)$, it can be derived from \cite[Lemma 2.3]{lipraeger1} easily.

 \begin{lemma}\label{pointstab}{\rm(\cite[Lemma 2.3]{lipraeger1})}
 1. Let $(\K_n, \mathcal P)$ be a circulant $(M,G)$-homogeneous factorisation. Then $\P(1)$ is a partition of $Z_n\setminus \{1\}$ and each factor digraph $\G_i=\Cay(Z_n,P_i(1))$. Let $G_1$ be the point stabilizer. Then  $G_1$ induces a transitive action on $\P(1)$, $G_1^{\P}=G^{\P}$ and the $G_1$-actions on $\P$ and $\P(1)$
 are equivalent.

 2.  Let $\P(1)=\{P_1(1),\ldots,P_k(1)\}$ be  a partition of $Z_n\setminus \{1\}$. Define $\G_i=\Cay(Z_n, P_i(1))$ and let $P_i$ be the set of arcs of $\G_i$.  Then $ \P=\{P_1,\ldots,P_k\}$ is a partition of the arc set of $\K_n$. Let $G$ be such that $\hat Z_n\le G\le \Sym(Z_n)$. Suppose the point stabilizer $G_1$  leaves $\P(1)$ invariant, and acts transitively on $\P(1)$. Then $G=\hat Z_nG_1$  leaves $\P$ invariant, and acts transitively on $\P$. Let $M$ be the kernel of the action of $G$ on $\P$. Then $(\K_n, \mathcal P)$ is a circulant $(M,G)$-homogeneous factorisation.
\end{lemma}


In the papers \cite{lipraeger, pls}, circulant homogeneous factorisations of complete digraphs $\K_n$ have been studied.   The following theorem gives some   basic properties of such factorisations. %we denote by $\K (B)$ the complete graph with vertex set $B$.


\begin{theorem}\label{lpthm} {\rm(\cite[Theorem 4.1]{lipraeger})}
Let $(\K_n, \mathcal P)$ be a circulant $(M,G)$-homogeneous factorisation of index $k$. Then the following statements hold:

(i) $G$ is soluble and $G^{\mathcal P}\cong \Z_k$.


(ii) for each prime divisor $r$ of $n$, $k|(r-1)$.
\end{theorem}

 We list some results concerning the induced sub-factorisations  in the following lemma,   the proofs  of these results can be found in \cite{lipraeger,lsx}.
\begin{lemma}\label{block}
 Let $(\K_{n}, \P)$     be a  circulant $(M, G)$-homogeneous factorisation  of index $k$ where $\P=\{P_1,\ldots,P_k\}$ and  identify the vertex set of $\K_{n}$ with the regular group $Z_{n}$.

 (1) Let $\D$ be a block of $G$ with order $m$ where $m|n$. Then $\D=xZ_{m}$ is a coset of the subgroup $Z_{m}$, and  the induced sub-factorisation $(\D,\P_\D)$ is a  circulant $(M_\D^\D, G_\D^\D)$-homogeneous factorisation of index $k$.

(2) Let $\s\in G$ be a cyclic isomorphism which fixes the point 1 and maps $P_i\to P_{i+1}$ (reading the subscript modulo $k$). Let $\D= Z_{m}$ be a block of $G$. Then $\s$ fixes $Z_m$ setwise and $\s|_{Z_m}$ is a cyclic isomorphism of the induced sub-factorisation $(\D,\P_\D)$ where $\P_{\D}=\{P_1^{\D}, P_2^{\D},\ldots,P_k^{\D}\}$.  Moreover $\s|_{Z_m}: P_i^{\D}\to P_{i+1}^{\D}$.

(3) Let $B$ be a minimal
block of $G$ such that $1\in B$. Then $B=Z_p$ for some prime $p$ and
$ \hat Z_p\lhd G_{Z_p}^{Z_p}\le \AGL(1,p)$.

 (4) Suppose $n=p^d (d\ge 2)$ is an odd prime power.
   Then $Z_{p}\subset Z_{p^2}\subset\cdots \subset Z_{p^{d-1}}$ is a block chain of $G$. Let $G_1$ be the point stabilizer of $1$. Then $G_1$ maps elements of order $p^i$ to elements of order $p^i$ for $i=1,\ldots,d$.
  \end{lemma}
\begin{proof}  (1) Let $B$ be a   block of $G$ such that $1\in B$ and $|B|=m$.
 Consider the   multiplications by the
elements in $B$, we have $BB=B$. Thus $B=Z_m$ is the subgroup of order $m$ of $Z_n$.
Suppose next that $\D$ is a block of order $m$, and $x\in \D$. Then $\D=xZ_{m}$ as required.
Hence $M_\D^\D$ contains the   regular subgroup $Z_m$. It follows from Lemma \ref{lpblock} that the   induced sub-factorisation $(\D,\P_\D)$ is a circulant $(M_\D^\D, G_\D^\D)$-homogeneous factorisation of index $k$.

(2) Since $\s$ fixes $1$ and $1\in \D$, $\s$ fixes the block $\D$. Since
$P_i^{\D}=P_i\cap (\D\times \D)$, $\s|_{\D}:P_i^{\D}\to
 P_{i+1}^{\D}$.  Therefore $\s|_{Z_m}$ is a cyclic isomorphism of the induced sub-factorisation $(\D,\P_\D)$.

(3)  This is  \cite[Lemma 4.4]{lipraeger}.

(4) It follows from \cite[Lemma 2.4]{lipraeger} and Corollary \ref{X-prim} that $Z_{p}\subset Z_{p^2}\subset\cdots \subset Z_{p^{d-1}}$ is a block chain of $G$. For $i=1,\ldots,d$, $G_1$ fixes block $Z_{p^i}$ and so maps elements of order $p^i$ to elements of order $p^i$. (See also \cite[Lemma 4.3, Corollary 4.4]{lsx}.)
\end{proof}



 We also need the following useful lemma. Since  the circulant homogeneous factorisation must be  cyclic by Theorem \ref{lpthm} (1),  the following lemma is a direct corollary of \cite[Lemma 5.2]{lipraeger1}.

\begin{lemma}\label{unique}{\rm(\cite[Lemma 5.2]{lipraeger1})}  Let $(\K_{n}, \P)$  be a  circulant $(M,G)$-homogeneous factorisation  of index $k$. Let $K$ be the kernel of the $G$-action on $\P$, and let $\mathcal B$ be a nontrivial $G$-invariant partition of $V(\K_n)$. Then each element of $G\setminus K$ fixes exactly one block of $\mathcal B$.
\end{lemma}

We finish this section by giving the following easy proposition.
\begin{prop}\label{d=1} Suppose that    $\F=(\K_{p},\P)$ is a  circulant homogeneous factorisation of order $p$ where $p$ is an odd prime. Then $\F$ is normal.
\end{prop}
\begin{proof}  Let $G=\Aut(\F)$ and let $M$  be the  kernel  of $G$ acting on $\P$. By Theorem \ref{lpthm} (1) $G$ is solvable, and hence $G$ is a primitive solvable c-group.  By Corollary  \ref{X-prim}, $\hat Z_{p}\lhd G\le \hat Z_p\rtimes Z_{p-1}$. Any element $\s\in G\setminus M$ is a cyclic isomorphism of $\F$ and hence $\F$ is normal by definition.
\end{proof}
\section{Lexicographic product constructions}
  Suppose $(\K_n,\P, X, Y)$ is a circulant  homogeneous factorisation of index $k$ and let $\s\in Y$ be a cyclic isomorphism. Recall that we also refer to $(\K_n, \P, X, \s)$ as this factorisation.
\begin{lemma}\label{lexi}Suppose that $\mathcal F_i=(\K_{n_i}, \P_i,  M_i, \s_i) (i=1,2)$  are two circulant homogeneous factorisations of index $k$ where $\s_i$  are cyclic isomorphisms of $\F_i$.  Let $\P_i=\{P_{i,1},\ldots,P_{i,k}\}$  and suppose that $\s_i: P_{i,j}\mapsto P_{i,j+1}$
(reading the second subscript modulo $k$). Let
 $\F_1[\F_2] =(\K_{n_1 n_2},\P, M, \s_1\times \s_2)$ where $\P=\{P_{1,j}[P_{2,j}]| 1\le j\le k\}$ and $M=M_2\wr M_1$.   Then $\F_1[\F_2]$ is a circulant homogeneous factorisation of index $k$ and $Z_{n_2}$ is a block of $\la M, \s_1\times \s_2\ra $.
Moreover, for any $j\in \{1,\ldots,k\}$, let  $\Cay(Z_{n_1n_2}, S_j)$ be the factor digraphs of $\F_1[\F_2]$. Then $sZ_{n_2}\subseteq S_j$ for any $s\in S_j\setminus Z_{n_2}$.
\end{lemma}


 \begin{proof} Write the vertex set $\K_{n_1n_2}$ as $V(\K_{n_1})\times V(\K_{n_2})$.
 Then it is easy to see that $\P=\{P_{1,j}[P_{2,j}]| 1\le j\le k\}$ is a partition of the arc set of $\K_{n_1n_2}$.

Let $\G_{i,j} (i=1,2, j=1,\ldots,k)$ be the factors of $\F_i$. Then $
 \G_{1,j}[\G_{2,j}]$ are factors of $(\K_{n_1n_2}, \P)$. For each $j=1,\ldots.,k$, since $M_1\le \Aut (\G_{1,j}), M_2\le \Aut(\G_{2,j})$, the automorphism group of $\G_{1,j}[\G_{2,j}]$ contains $M_2\wr M_1$ which is transitive on the vertex set. In addition, let $\la x_i\ra\subseteq M_i$ be the corresponding regular cyclic group on $V(\K_{n_i})$. Then $(1,\cdots, 1, x_2)x_1(\in M_2\wr M_1)$ generates a regular cyclic group on $V(\K_{n_1n_2})$. Lastly, it is easy to check  that the natural action of $\s_1\times\s_2$ on $V(\K_{n_1})\times V(\K_{n_2})$ maps $P_{1,j}[P_{2,j}]$ to  $P_{1,j+1}[P_{2,j+1}]$. Therefore $(\K_{n_1n_2}, \P)$ is a circulant $(M, \s_1\times \s_2)$ homogeneous factorisation of index $k$.

We may assume that   the factor digraph $\G_{1,j}[\G_{2,j}]$ is a circulant    $\Cay(Z_{n_1n_2}, S_j)$ for $ j=1,\ldots,k$.  For  any $u\in V(\K_{n_1})$, let $B_u=\{(u,v)| v\in  V(\K_{n_2})\}$. Then $\{B_u| u\in V(\K_{n_1})\}$ is a block system of $\la M_2\wr M_1,  \s_1\times \s_2 \ra$ acting on the vertex set $V(\K_{n_1})\times V(\K_{n_2})$. By Lemma \ref{block} (1), $\{B_u| u\in V(\K_{n_1})\}=\{xZ_{n_2}|x\in Z_{n_1n_2}\}$, hence it is easy to deduce that $sZ_{n_2}\subseteq S_j $ for any $s\in S_j\setminus Z_{n_2}$.
\end{proof}

 \begin{lemma}\label{lexi2} Let $n, n_1, n_2$ be   positive integers   such that $n=n_1n_2$ and $n_1, n_2\ge 2$. Let $\F=(\K_{n},\P, X, \s)$ be a circulant homogeneous factorisation of index $k$ where $\s$ is a cyclic isomorphism fixing point 1  and let $\P=\{P_1,\ldots,P_k\}$.
 Let $\si_j=\Cay (Z_{n}, P_j(1)) (j=1,\ldots,k)$ be the corresponding factor digraphs. Suppose that $Z_{n_2}$ is a block of $Y=\la X,\s\ra$ and for any $j\in \{1,\ldots,k\}$ and any $s\in P_j(1) \setminus Z_{n_2}$,   $sZ_{n_2}\subseteq P_j(1)$.
Then there exists two circulant homogeneous factorisations $\F_i$ of order $n_i, (i=1,2)$ such that $\F=\F_1[\F_2]$.
 \end{lemma}

\begin{proof}

Let $\F_2=(\K_{n_2}, \P_2, X_{Z_{n_2}}^{Z_{n_2}}, Y_{Z_{n_2}}^{Z_{n_2}})$ be the induced sub-factorisation  on the block $Z_{n_2}$ by $\F$ where $\P_2=\{P_{2,j}|j=1,\ldots,k\}$ and $P_{2,j}=P_j\cap (Z_{n_2}\times Z_{n_2})$.
 By Lemma \ref{block} (1) (2), $\F_2$ is a circulant homogeneous factorisation of order $n_2$ and $\s|_{Z_{n_2}}$ is a cyclic isomorphism of $\F_2$.


Let $\bar X$ and $\la\bar \s\ra$ be the induced permutation groups  of $X$  and $\la \s \ra$  on the block system $Z_{n}/Z_{n_2}=\{gZ_{n_2}  |g\in Z_{n}\}$ respectively. Then $\widehat{Z_n/Z_{n_2}}\le \bar X$.
 Next we  define an induced quotient factorisation $\F_1$ on this block system $Z_{n}/Z_{n_2}$.  For $j=1,\ldots,k$, let
$$P_{1,j}(1)=  P_j(1)\setminus Z_{n_2}   \mbox{ and }P_{2,j}(1)=  P_j(1) \cap Z_{n_2}.$$ Then $P_j(1)=P_{1,j}(1)\cup P_{2,j}(1)$. First note that $\G_{2,j}:=\Cay(Z_{n_2},P_{2,j}(1)) (j=1,\ldots,k)$ are the factor digraphs of $\F_2$. Suppose that $\s: P_j\to P_{j+1}$. Then $\s|_{Z_{n_2}}:P_{2,j}(1)\to P_{2,j+1}(1)$ and hence  $\s|_{Z_{n_2}}:\G_{2,j}\to \G_{2,j+1}$.

On the other hand,
  it is proved in \cite[Lemma 2.2]{l6} that
$\Cay(Z_{n},P_{1,j}(1))=\G_{1,j}[\ov {\K_{n_2}}]$ where $\G_{1,j}=\Cay(Z_{n}/Z_{n_2},\ov {P_{1,j}(1)})$ and $\ov {P_{1,j}(1)}=\{\ov s=sZ_{n_2}\mid s\in P_{1,j}(1)\}$.
 Then $\bar \s: \ov {P_{1,j}(1)} \to \ov {P_{1,j+1}(1)}$ and so $\bar \s:  \G_{1,j}\to \G_{1,j+1}$.
 Moreover, $\si_j=\G_{1,j}[\G_{2,j}]$ is a
lexicographic product graph.   Let $\P_1=\{P_{1,j}|j=1,\ldots,k\}$ where $P_{1,j}$ is the arc set  of $\G_{1,j}$. Write the vertex set of $\K_{n_1}$ as the quotient group $Z_{n}/Z_{n_2}$ and let $\F_1=(\K_{n_1}, \P_1)$.
 It is easy to deduce that $\F_1$ is a circulant $(\bar X,\bar \s)$-homogeneous factorisation of order $n_1$ and $\F=\F_1[\F_2]$
as required.
\end{proof}



\begin{corollary}\label{lexico}   Let $\F=(\K_{p^d},\P) $ be a circulant homogeneous factorisation of index $k$ where $p^d$ is an odd prime power and let $m<d$ be a positive integer. For any $j\in \{1,\ldots,k\}$, let  $\Cay(Z_{p^d}, S_j)$ be the factor digraphs of $\F$. Then the following two statements are equivalent.

1. There exist  two circulant homogeneous factorisations $\F_i  (i=1,2)$,   such that $\F=\F_1[\F_2]$  where $\F_2$ is of order $p^m$.

 2. For any $j\in \{1,\ldots,k\}$ and any $s\in S_j$ with $o(s)> p^m$, $sZ_{p^m}\subseteq S_j$.
\end{corollary}

\begin{proof} Just note that by Lemma \ref{block} (4), $Z_{p^m}$
is a block of $\Aut(\F)$, the result follows from Lemma \ref{lexi} and \ref{lexi2}. \end{proof}
  Moreover  we have the following useful remark.

\begin{remark}\label{lexire}Let $\F=(\K_{n},\P, X, \s)$ be a circulant homogeneous factorisation of index $k$ where $\s$ is a cyclic isomorphism fixing point 1  and let $\P=\{P_1,\ldots,P_k\}$. Suppose further that $\F=\F_1[\F_2]$ is of lexicographic product form where $\F_2$ is of order $n_2$, and $Z_{n_2}$ is a block of $\la X,\s\ra$. Then $\F_2$ is the    sub-factorisation induced on the block $Z_{n_2}$ by $\F$, and $\F_1$ is the quotient factorisation induced on the block system $\{gZ_{n_2}| g\in Z_{n}\}$ defined as in the proof of Lemma \ref{lexi2}. In particular, with the notation in Lemma \ref{lexi2}, the factor digraphs of $\F_1$ are the quotient cayley digraphs $\G_{1,j}=\Cay(Z_{n}/Z_{n_2},\ov {P_{1,j}(1)})$ and $\ov {P_{1,j}(1)}=\{\ov s=sZ_{n_2}\mid s\in P_{1,j}(1)\}$. Suppose that $\s: P_j\to P_{j+1}$. Then $\bar \s: \G_{1,j}\to \G_{1,j+1}$.

\end{remark}




\section{Cyclic isomorphisms and normal circulant homogeneous factorisations of order $p^d$ }


Let $(\K_{p^d},\P)$ be a circulant $(X,Y)$-homogeneous factorisation of index $k$, and suppose that $X$ contains a regular cyclic subgroup $Z_{p^d}$. We always identify the vertex set of $\K_{p^d}$ with the group $Z_{p^d}$. The following lemma discusses
the orders of  the cyclic isomorphisms.

\begin{lemma}\label{orders}Let $\F=(\K_{p^d},\P)$ be a circulant $(X, Y)$-homogeneous factorisation of index $k$ and let $\t$
 be a  cyclic isomorphism.
  Then $k|(p-1)$ and $k|o(\t)$. Moreover  there exists a cyclic isomorphism $\s\in Y$
     such that  $\s$ fixes 1 and $r|k$ for each prime divisor $r$ of the order $o(\s)$.
 \end{lemma}
\begin{proof} By Theorem \ref{lpthm}, $k|(p-1)$. Since $\t$ is transitive on $\P$ and $|\P|=k$, $k|o(\t)$. By Theorem \ref{lpthm} again,  there exists a cyclic isomorphism $\s\in Y\setminus X$ such that $Y=\la X,\s\ra$ and $Y^{\P}=\la \s ^{\P}\ra$ is a cyclic group of order $k$.
Since $\hat Z_{p^d}\le X$ is vertex transitive, we may assume that $\s$ fixes the point 1.  Let $q$ be a prime such that $(q,k)=1$. For any positive integer $m$, $\s^{q^m}(\in Y)$ is still transitive on $\P$ and so is also a cyclic isomorphism of $\F$. Therefore replacing $\s$ by some power of $\s$ if necessary, we may assume $r|k$ for each prime divisor $r$ of the order $o(\s)$.
\end{proof}

\begin{remark}\label{ordersre}

Let $\F=(\K_{p^d},\P)$ be a circulant  homogeneous factorisation of index $k$.
For convenience we will assume from now on that, for a cyclic isomorphism $\s$ of $\F$, $\s$ fixes the point 1. Moreover  if $r|k$ for each prime divisor $r$ of the order $o(\s)$, then we say the cyclic isomorphism $\s$ \emph{satisfies Lemma \ref{orders}}.
\end{remark}


Suppose further that the circulant  homogeneous factorisation   $(\K_{p^d},\P)$ is normal, that is  there exists a cyclic isomorphism $\s$  normalising the regular subgroup $Z_{p^d}$.   Hence $\s$ can be viewed as an
automorphism of the group $Z_{p^d}$, that is $\s
\in \Aut(Z_{p^d}) \cong  Z_{p^{d-1}(p-1)}$.

\begin{construction}\label{con1}
Let $p^d$ be an odd prime power such that $d\ge 1$ and $k\ge 2$ a positive integer such that $k|(p-1)$.
Suppose that $\s\in \Aut(Z_{p^d})$ such that each prime divisor of $o(\s)$ divides $k$ and $k|o(\s)$.   In particular $(o(\s), p)=1$.  Since $\Aut(Z_{p^d}) \cong  Z_{p^{d-1}}\times Z_{p-1}$,   $\s$  belongs to the cyclic subgroup of order $p-1$ and
   hence  $\s$   acts semiregularly on $Z_{p^d}\setminus\{1\}$. Let $\D_1,\ldots,
\D_t$ be the orbits of  $\s$ on  $Z_{p^d}\setminus\{1\}$. As  $k|o(\s)$, $\s^k$ divides each $\D_i$ into $k$ orbits, say $\D_{i,1},\D_{i,2},\ldots,\D_{i,k}$. Relabeling if necessary, we may assume that $\D_{i,j}^{\s}=\D_{i,j+1}$ where $i\in\{1,\ldots,t\}, j\in\{1,\ldots,k\}$ (reading the second subscript modulo $k$). For any $j_1,\ldots,j_t\in \{1,\ldots,k\}$,   let
$$P_1(1)=  \D_{1,j_1}\cup \D_{2,j_2}\cup\cdots\cup\D_{t, j_t}.$$ For $i=2,\ldots,k$, let
$$P_i(1)=P_1(1)^{\s^{i-1}}= \D_{1,j_1+(i-1)}\cup \D_{2,j_2+(i-1)}\cup\cdots\cup\D_{t, j_t+i-1}.$$ Then $\P(1) =\{P_1(1),\ldots,P_k(1) \}$ is a $\s$-invariant   partition of $Z_{p^d}\setminus\{1\}$.
For each $i$, let $\G_i =\Cay(Z_{p^d}, P_i(1))$, let $P_i $ be  the set of arcs of the circulant $\G_i $, and let $\P =\{P_1,\ldots,P_k\}$. By Lemma \ref{pointstab} (2) (taking $G$ as $\la Z_{p^d}, \s\ra$),   $\F=(\K_{p^d},\P )$    is a normal circulant homogeneous factorisation of index $k$ and $\s$ is a cyclic isomorphism of this factorisation. Denote $J=\{j_1,\ldots,j_t\}$. Since this construction $\F$    depends on the choice of $\s$ and $J$, we also denote this construction by $\F_{\s, J}$.
\end{construction}

Note that there do exist $p, d, k,\s$ satisfying the conditions of Construction \ref{con1}. Conversely, suppose that $(\K_{p^d},\P)$ is a normal circulant homogeneous factorisation of index $k$.  Then each factor digraph $\G_i$ is a circulant $\Cay(Z_{p^d}, P_i(1))$ where $P_i(1)=\{\a\in  Z_n| (1,\a)\in P_i\}$. Let $\s\in \Aut(Z_{p^d})$ be a cyclic isomorphism. As proved in  Lemma \ref{orders}, we may assume that each prime divisor  of the order $o(\s)$ divides $k$.  Therefore $\s$    induces a transitive action on $\P(1)=\{P_1(1),\ldots,P_k(1)\}$ and $\s^k$ fixes each $P_i(1)$ for $i=1,\ldots,k$.  As defined above in Construction \ref{con1}, let $\D_1,\ldots,
\D_t$ be the orbits of  $\s$ on  $Z_{p^d}\setminus\{1\}$. And for each $i$,   $\s^k$ divides  $\D_i$ into $k$ orbits, say $\D_{i,1},\D_{i,2},\ldots,\D_{i,k}$ such that $\D_{i,j}^{\s}=\D_{i,j+1}$.     It is easy to show that
there exists $j_1,\ldots,j_t\in \{1,\ldots,k\}$  such that
 $P_1(1)= \{\D_{1,j_1}\cup \D_{2,j_2}\cup\cdots\cup\D_{t, j_t}\}$ and
 $P_i(1)=P_1(1)^{\s^{i-1}}$ as in Construction \ref{con1}.
Therefore  Construction~\ref{con1} provides us with a method for constructing
all  normal circulant homogeneous factorisations of order $p^d$, we write this result in the following proposition.

\begin{prop}\label{normalcon}Let $\F$ be a normal circulant homogeneous factorisations of order $p^d$ and index $k$. And let $\s\in \Aut(Z_{p^d})$ be a cyclic isomorphism of $\F$ which satisfies Lemma \ref{orders}. With above notation, there exists $J=\{j_1,\ldots,j_t\}$ such that $\F=\F_{\s,J}$ which is defined  in Construction \ref{con1}.
\end{prop}
We will need the following easy lemma for the proof of the main theorem.
\begin{lemma}\label{auto} Let $p^d$ be an odd prime power such that $d\ge 2$, and let $\s\in \Aut(Z_{p^d})$ such that $o(\s)|(p-1)$. Let $Z_{p^i}\le Z_{p^d}$ be a subgroup. Then

(1) $o(\s|_{Z_{p^i}})=o(\s)$.

(2) Let $\bar \s \in \Aut(Z_{p^d}/Z_{p^i})$ be the automorphism of the quotient group induced by $\s$. Then $o(\s)=o(\bar \s)$.
  \end{lemma}

  \begin{proof}(1)  Suppose $\Aut(Z_{p^d})=\la \mu \ra\times \la \g\ra=Z_{p-1}\times
Z_{p^{d-1}}$. Then $\mu$ is a product of $\frac{p^d-1}{p-1}$ disjoint $(p-1)$-cycles acting on $Z_{p^d}\setminus \{1\}$.  By assumption $\s \in \la\mu\ra$ and hence  $o(\s|_{Z_{p^i}})=o(\s)$.

(2) It is easy to check that $\Aut(Z_{p^d}/Z_{p^i})= \Aut(Z_{p^d})/\la
\g^{p^{d-1-i}}\ra$ and so $o(\s)=o(\bar\s)$.
  \end{proof}
\begin{lemma}\label{use}
Suppose that $(\K_{p^d},\P)$ is a normal circulant homogeneous factorisation of index $k$ and   $\s\in \Aut(Z_{p^d})$ is a cyclic isomorphism satisfying Lemma \ref{orders}. Let $\t \in Aut(Z_{p^d})$ such that $k|o(\t)$ and each prime divisor of $o(\t)$ divides $k$. Then $\t$ is a  cyclic isomorphism of $(\K_{p^d},\P)$ if and only if  $o(\sigma)=o(\t)$.
 \end{lemma}

\begin{proof}
 By assumption  we may assume that $o(\s)=mk, o(\t)=nk$ where $m, n$ are positive integers and each prime divisor of $m$ and $n$ divides $k$.

 Suppose first that $o(\s)=o(\t)$.
 Since $\Aut(Z_{p^d})$ is cyclic, $\t=\s^l$ for some integer $l$ such that  $(l, o(\t))=1$. This implies $(l,k)=1$. Since $\s$ is transitive on $\P$ and $|\P|=k$, so is $\t$. Therefore   $\t$ is a cyclic isomorphism of $(\K_{p^d},\P)$.



Conversely, suppose   that $\t$ is a cyclic isomorphism. Then $\t$ is transitive on $\P$. Let $H=\la\s,\t\ra$. Then $H$ is transitive on $\P$.
 Note that $\s, \t\in \Aut(Z_{p^d})\cong Z_{(p-1)p^{d-1}}$, we have that $H=\la\a\ra$   where $\a$ generates $H$ and   $o(\a)=\frac{mk\cdot nk}{(m,n)k}=\frac{mn}{(m,n)}k$.   Suppose that $m\ne n$, without loss of generality, we may assume  $\frac{m}{(m,n)}\ne 1$ and   $\t=\a^{\frac{m}{(m,n)}}$. Since both $\a$ and $\t$ are transitive on $\P$, $(k,\frac{m}{(m,n)})=1$. However by assumption each prime divisor of $m$ divides $k$, a contradiction. Therefore $m=n$ and $o(\s)=o(\t)$.
\end{proof}


\begin{lemma}\label{normalcyc}Suppose that $\F=(\K_{p^d},\P)$ is a normal circulant homogeneous factorisation of index $k$ where $\P=\{P_1,..,P_k\}$. Let $\s$ be a cyclic isomorphism (not required to be a group automorphism) such that $\s: P_i\to P_{i+1}$  (reading the subscript modulo $k$). Then there exists a cyclic isomorphism $\t\in \Aut(Z_{p^d})$
 satisfying Lemma \ref{orders} and $\t: P_i\to P_{i+1}$ too.
\end{lemma}
\begin{proof}
Since $\F$ is normal, by definition there exists a cyclic isomorphism $\t\in \Aut(Z_{p^d})$
 satisfying Lemma \ref{orders}. Consider $Y=\la \hat Z_{p^d}, \s,\t\ra$ and let $X$ be the kernel of $Y$ acting on $\P$. Let $\bar \s=\s^{\P}$ and $\bar \t=\t^{\P}$.   Then $Y/X=\la \bar\s \ra=\la\bar\t\ra$, and hence $\bar\s=\bar\t^l$ for some positive integer. Replacing $\t$ by $\t^l$, we have that $\t: P_i\to P_{i+1}$ as required.
\end{proof}

Lastly in this section we give examples of non-normal circulant homogeneous factorisations of order $p^2$. Note that there do exist $p, k, \s_1, \s_2$ satisfying the conditions of the following proposition.

\begin{prop}\label{nonnormal}Let $p$ be an odd prime  and $k(\ge 2)$ a positive integer such that $k|(p-1)$.
Suppose further that $\s_1, \s_2\in \Aut(Z_{p})$ such that  $o(\s_1)=k$ and $o(\s_2)=mk$ where $m\ge 2$ and each prime divisor of $m$ divides $k$. By Construction \ref{con1}, there exist  circulant homogeneous factorisations $\F_i $ of index $k$ and order $p$ such that  $\s_i$ is a cyclic isomorphism of  $\F_i$ respectively. Take $\F=\F_1[\F_2]$. Then $\F$ is  a non-normal circulant homogenous factorisation of order $p^2$.
\end{prop}
\begin{proof} By Lemma \ref{lexi} $\F$ is a circulant homogeneous factorisation of order $p^2$. Suppose conversely that $\F$ is normal and $\s \in \Aut(Z_{p^2})$ is a cyclic isomorphism of $\F$ satisfying Lemma \ref{orders}.  By Remark \ref{lexire},  $\F_2$ is the sub-factorisation induced on the  block $Z_p$. By Lemma \ref{block} (2), $\s|_{Z_{p}}\in \Aut(Z_{p})$
is also a cyclic isomorphism of $\F_2$. By Lemma \ref{use}, $o(\s|_{Z_{p}})=o(\s_2)$. It follows from Lemma \ref{auto} that $o(\s_2)=o(\s)$.  On the other hand, $\F_1$ is the quotient factorisation of the block system $\{gZ_p|g\in Z_{p^2}\}$ by Remark \ref{lexire}, and the induced $\bar \s\in \Aut(Z_{p^2}/Z_{p})$ is also a cyclic isomorphism of $\F_1$. Still by Lemma \ref{use} and Lemma \ref{auto}, we have $o(\s_1)=o(\bar\s)=o(\s)$ which contradicts the fact that $o(\s_1)\ne o(\s_2)$. Therefore $\F$ is not normal.
\end{proof}

\section{Proof of Theorem \ref{mainthm}}
  We first study the structures of circulant homogeneous  factorisations of order $p^d$.

\begin{lemma}\label{step1} Let $(\K_{p^d},\P, X, Y)$ be a circulant homogeneous factorisation of index $k$, and let $\s\in Y$ be a cyclic isomorphism.  Let $\G_i=\Cay(Z_{p^d}, S_i)$ be the factor digraphs where $1\le i\le k$. Then   either $\hat Z_{p^d}\lhd Y$
 and $\s\in\Aut(Z_{p^d})$, or for any $i\in \{1,\ldots,k\}$ and any $s\in S_i$ with
 $o(s)=p^d$,  $sZ_{p}\subset S_i$.
\end{lemma}
\begin{proof} If $d=1$ then $Y$ is a soluble primitive c-group. It follows from
Corollary \ref{X-prim} that $\hat Z_{p}\lhd Y\le \AGL(1,p)$ and $\s\in \Aut(Z_p)$.

Next suppose that $d\ge 2.$ Let  $X_1$
and $Y_1$ be the point
 stabilizers of $1$ in $X$ and $Y$ respectively. Then $Y_1=\la X_1, \s\ra$ and suppose that $\s: S_i\to S_{i+1} (1\le i\le k)$, reading the subscript modulo $k$.

 First take any $g\in S_1$ with $o(g)=p^d$. Consider the
orbital digraph $\G_g$ of $Y$ with   arc set $\D=(1,g)^Y$. Then
 $\G_g=\Cay(Z_{p^d},S_{1g}\cup S_{2g}\cup\cdots \cup S_{kg})$ where
 $S_{1g}=g^{X_1}$ is a subset of $S_1$ and  $S_{ig}=(S_{1g})^{\s^{i-1}}\subseteq S_i$ for $i=2,\ldots,k$. Since
$o(g)=p^d$, $\G_g$ is a connected $Y$-arc-transitive circulant of
order $p^d$.
By Lemma \ref{block} (4),   $o(h)=p^d$ for any $h\in S_{1g}\cup S_{2g}\cup\cdots \cup S_{kg}$.
In particular, $\G_g$ is not the complete graph $\K_{p^d}$.
 Applying Theorem \ref{arc-circ}, we have either
  $\hat{Z_{p^d}}\lhd Y$ and $\s\in \Aut(Z_{p^d})$ or
$gZ_{p}\subset S_{1g}\cup S_{2g}\cup\cdots \cup S_{kg}$.


In the latter case, we claim that $gZ_p\subset S_{1g}\subset S_1$. Note that $\mathcal B=\{xZ_p\mid x\in
 Z_{p^d}\}$ is a block system of $Y$.  Suppose $gZ_p\cap
 S_{ig}\not=\varnothing$ where $k\ge i\ge 2$. Then $g^{\s^{1-i}} Z_p\cap
 S_{1g}\not=\varnothing$. Since $S_{1g}=g^{X_1}$,   there exists $x\in X_1$ such that $g^x\in g^{\s^{1-i}}
 Z_p$. It follows that $(gZ_p)^{\s^{1-i}}=(gZ_p)^x$ and so $gZ_p$ is fixed by $x\s^{i-1}$. Since $gZ_p\ne Z_p$,  $x\s^{i-1}$ fixes at least two blocks $gZ_p$ and $Z_p$,
contradicting Lemma \ref{unique}. Hence $gZ_p\subset
 S_{1g}$.

 For any $g_i\in S_i (i\ge 2)$ with $o(g_i)=p^d$, we consider the
orbital digraph $\G_{g_i}$ of $Y$ with   arc set $\D=(1,g_i)^Y$ as well.  Applying the same argument as above repeatedly, it is easy to deduce that either $\hat Z_{p^d}\lhd Y$
 and $\s\in\Aut(Z_{p^d})$, or for any $i\in \{1,\ldots,k\}$, $sZ_{p}\subset S_i$ where $s\in S_i$ with
 $o(s)=p^d$.
\end{proof}


\begin{prop}\label{s sets} Let $(\K_{p^d},\P, X, Y)$ be a circulant homogeneous factorisation of index $k$, and let $\s\in Y$ be a cyclic isomorphism.  Let $\G_i=\Cay(Z_{p^d}, S_i)$ be the factor digraphs. Then  there
exists $n\in \{1,\ldots,d\}$ such that  $\hat{Z_{p^n}}\lhd Y_{Z_{p^n}}^{Z_{p^n}}$ and
$\s|_{Z_{p^n}}\in\Aut(Z_{p^n})$.  Moreover, for each $i\in \{1,\ldots,k\}$ and each $s\in S_i$ with
$o(s)>p^n$, we have $sZ_p\subset S_i$.
\end{prop}
\begin{proof}
 If $d=1$, then take $n=1$, the result follows from Corollary \ref{X-prim}. Suppose next that $d\ge 2$.
 By Lemma \ref{step1}, we
may assume   that for any $i\in \{1,\ldots,k\}$,   $sZ_p\subset S_i$ where $s\in S_i$ with    $o(s)=p^d$.  By Lemma \ref{block} (4), the subgroup $Z_{p^{d-1}}$ is a block of $Y$
and the induced factorisation $\F_{d-1}=(\K_{p^{d-1}}, \P_{Z_{p^{d-1}}}, X_{Z_{p^{d-1}}}^{Z_{p^{d-1}}}, Y_{Z_{p^{d-1}}}^{Z_{p^{d-1}}})$  is a circulant homogeneous factorisation where $Y_{Z_{p^{d-1}}}^{Z_{p^{d-1}}}=\la X_{Z_{p^{d-1}}}^{Z_{p^{d-1}}}, \s|_{Z_{p^{d-1}}}\ra$. For any $i\in \{1,\ldots,k\}$, let $\G_i(d-1)=\Cay(Z_{p^{d-1}}, S_i(d-1))$ be the corresponding factor digraphs of $\F_{d-1}$.
Then $S_i(d-1)=\{s\in S_i | o(s)\le
 p^{d-1}\}$.
  Applying
  Lemma \ref{step1} to $\F_{d-1}$, we deduce that
  either $\hat{Z_{p^{d-1}}}\lhd Y_{Z_{p^{d-1}}}^{Z_{p^{d-1}}}$
   and $\s|_{Z_{p^{d-1}}}\in \Aut(Z_{p^{d-1}})$,
  or   for any $i\in \{1,\ldots,k\}$,   $sZ_p\subset S_i$ where $s\in S_i$ with  $o(s)=p^{d-1}$.

Continuing in this fashion (note that $\hat{Z_p}\lhd Y_{Z_p}^{Z_p}$
by Lemma \ref{block} (3)) we have that   an integer $n$ exists such that
$n\in\{1,\ldots,d\}$, $\hat{Z_{p^n}}\lhd Y_{Z_{p^n}}^{Z_{p^n}},$
$\s|_{Z_{p^n}}\in\Aut(Z_{p^n})$  and for any $i\in \{1,\ldots,k\}$, $sZ_{p}\subset S_i$ where $s\in S_i$ with
 $o(s)>p^n$.
\end{proof}

 Suppose $(\K_{p^d},\P, X, Y)$ is a circulant homogeneous factorisation and denote the factor digraphs by $\G_i=\Cay(Z_{p^d}, S_i)$. By Proposition \ref{s sets},   there exists $n\in \{1,\ldots,d\}$ such
that   $\hat{Z_{p^n}}\lhd Y_{Z_{p^n}}^{Z_{p^n}}$ and for each $i\in \{1,\ldots,k\}$ and each $s\in S_i$ with
$o(s)>p^n$, $sZ_p\subset S_i$.   Thus for any $i\in \{1,\ldots,k\}$, we divide the Cayley subset $S_i$
into the following two parts. Let
\begin{equation}\label{s1s2}S_i^1=\{s\in S_i\mid   o(s)\le p^n   \}  \mbox{ and }
S_i^2=\{s\in S_i\mid p^d\ge o(s)\ge p^{n+1} \}.\end{equation}   Then
$S_i=S_i^1\cup S_i^2$. Note that  we set $S_i^2=\varnothing$ if $n=d$.




\begin{lemma}\label{induction}Suppose that $\F=(\K_{p^d},\P, X, Y)$ is a circulant homogeneous factorisation of index $k$ with $d\ge 2$, and let $\s\in Y$ be a cyclic isomorphism.  For any $i\in\{1,\ldots,k\}$, let $\G_i=\Cay(Z_{p^d}, S_i)$ be the factor digraphs, and suppose that $\s: \G_i\to \G_{i+1}$ (reading the subscript modulo $k$).
 Suppose further that $n, S_i^1, S_i^2$ are defined as above in
$(\ref{s1s2})$. Then there exists a factorisation $\F'=(\K_{p^d}, \P')$ satisfying the following conditions.
\begin{itemize}
\item[(i)] $\F'$ is also a circulant $(X, Y)$-homogeneous factorisation of index $k$. Let $\si_i=\Cay(Z_{p^d}, T_i)$ be the corresponding
    factor digraphs of $\F'$. Then
 $\s: \si_i\to \si_{i+1}$ is  a cyclic isomorphism of $\F'$.

\item[(ii)]  For any $i\in \{1,\ldots,k\}$ and any $r\in T_i$ such that $o(r)>p$, we have $rZ_p\subseteq T_i$. (This implies $\F'$ is of lexicographic product form by Corollary \ref{lexico}.)


\item[(iii)] For any $i\in \{1,\ldots,k\}$, let $T_i^2=\{r\in T_i | o(r)>p^n\}$. Then $T_i^2=S_i^2$.

\item[(iv)] For any $i\in \{1,\ldots,k\}$, let  $S_i^0=\{s\in S_i \mid o(s)=p\}$ and   $ T_i^0=\{r\in T_i \mid o(r)=p\}.$ Then $T_i^0=S_i^0$.
\end{itemize}
\end{lemma}

\begin{proof}
If $n=1$, take $\F'=\F$. We next assume that $n\ge 2$.
Let $X_n=X_{Z_{p^n}}^{Z_{p^n}}, Y_n=Y_{Z_{p^n}}^{Z_{p^n}}$ and let $\t=\s|_{Z_{p^n}}$.  Then $Y_n=\la X_n, \t\ra$. By Lemma \ref{block} the induced sub-factorisation  (on the block $Z_{p^n}$) $\F_{n}=(Z_{p^n},\P_{Z_{p^n}})$ is $(X_n, Y_n)$-homogeneous and $\t$ is a cyclic isomorphism of $\F_n$.

 By Proposition \ref{s sets}, $X_n=\hat Z_{p^n}\rtimes L$ where $L\le\Aut(Z_{p^n})$ and fixes each $S_i^1 (1\le i\le k)$ setwise.
Moreover $\t\in \Aut(Z_{p^n})$ and $\t^k\in L$.
Applying Lemma \ref{block} to $\F_n$, $\mathcal B_1=\{gZ_p\mid
g\in Z_{p^n}\}$ forms a complete block system of $Y_n$ on
$Z_{p^n}$.  As $\t$ normalises $L$, $(gZ_p)^{L\t}=(gZ_p)^{\t L}$. For any $gZ_p\in\mathcal B_1$ such that $gZ_p\ne Z_p$, it follows from Lemma \ref{unique} that
$g^{\t^i}Z_p\ne (gZ_p)^x$ for any $x\in L$ and $1\le i<k$. Thus we may suppose that
$L$ has   $km$  orbits  (for some integer $m\ge 1$) on $\mathcal B_1\setminus \{Z_p\}$ and
after relabelling if necessary, we may suppose that
$\D_{j,1},\D_{j,2},\ldots,\D_{j,k}$ are  $L$-orbits on $\mathcal B_1\setminus \{Z_p\}$ such that $\D_{j,i}^{\t}=\D_{j,i+1}$ where $j\in\{1,\ldots,m\}, i\in\{1,\ldots,k\}$.

Let $$R_i= \bigcup_{j=1}^{m}\D_{j,i}\mbox{ and }
 S_i^0=\{s\in S_i^1\mid o(s)=p\}.$$  Take $T_i^1=S_i^0\cup R_i$ and take $T_i^2=S_i^2$. Let $T_i=T_i^1\cup T_i^2$. Then $\{T_1,\ldots,T_k\}$ is a partition of $Z_{p^d}\setminus \{1\}$.  By Lemma \ref{block} (4) and our construction it is easy to see that the vertex stabilizer $X_1$ of $X$ fixes each $T_i  (i=1,\ldots,k)$ and $\s: T_i\mapsto T_{i+1}$.
Define $\si_i=\Cay(Z_{p^d}, T_i), (i=1,\ldots,k)$ and let
$P_i'$ be the set of arcs of $\si_i$  and let $ \P'=\{P_1',\ldots,P_k'\}$. By Lemma \ref{pointstab} (2) $\F'=(\K_{p^d},\P')$
 is a  circulant $(X, Y)$-homogeneous factorisation of index $k$,
and $\s$ is  a cyclic isomorphism of $\F'$. The results (ii), (iii) and $(iv)$ follow from the construction of $\F'$ obviously.
\end{proof}


We are ready to prove   Theorem \ref{mainthm}.

\emph{Proof of Theorem \ref{mainthm}:} We proceed  by induction on
$d$. If $d=1$, the result follows from Proposition \ref{d=1}. Assume
inductively the result holds for  circulant homogeneous factorisations  of
order $\le p^{d-1}$ where $d\ge 2$.

Let $\F=(\K_{p^d},\P)$. Let $Y=\Aut(\F)$ and $X$ be the kernel of $Y$ acting on $\F$.
Then $\F$ is a circulant $(X,Y)$-homogeneous factorisation of index $k$. Let $\s$ be a cyclic isomorphism  satisfying Lemma \ref{orders} and  let $\G_i=\Cay(Z_{p^d}, S_i)$ be the factor digraphs $(1\le i\le k)$. We   assume that $\s : \G_i\to \G_{i+1}$  (reading the subscript modulo $k$).   By Proposition
\ref{s sets},  there
exists $n\in \{1,\ldots,d\}$ such that  $\hat{Z_{p^n}}\lhd Y_{Z_{p^n}}^{Z_{p^n}}$ and
$\s|_{Z_{p^n}}\in\Aut(Z_{p^n})$.   If $n=d$ then $\F$ is  normal as
required. So we assume next that $n<d$.

By Proposition \ref{s sets}, for each $i\in \{1,\ldots,k\}$ and each $s\in S_i$ with
$o(s)>p^n$, $sZ_p\subset S_i$.  Let $S_i^1=\{s\in S_i\mid   o(s)\le p^n   \}$   and
$S_i^2=\{s\in S_i\mid p^d\ge o(s)\ge p^{n+1} \}$.    Then
$S_i=S_i^1\cup S_i^2$.
 By Lemma \ref{induction}, there
exists   a circulant $(X,Y)$ homogeneous factorisation $\F'=(\K_{p^d}, \P')$ such that $\s$ is also a cyclic isomorphism of $\F'$.
Let $\si_i=\Cay(Z_{p^d}, T_i)$ be the corresponding
    factor digraphs of $\F'$. Then $\s: \si_i\to \si_{i+1},$  (reading the subscript modulo $k$). Setting  $T_i^1=\{s\in T_i\mid   o(s)\le p^n   \}$   and
$T_i^2=\{s\in T_i\mid p^d\ge o(s)\ge p^{n+1} \}$, we have $T_i^2=S_i^2$. Moreover, let  $S_i^0=\{s\in S_i \mid o(s)=p\}$ and $ T_i^0=\{r\in T_i \mid o(s)=p\}.$ Then $T_i^0=S_i^0$ by Lemma \ref{induction}.



 By Corollary \ref{lexico} $\F'$ is of lexicographic product form. Suppose $l(\ge 1)$ is maximal such that $\F'= \F'_1[\F'_2]$ where
$\F_2'$ is a   circulant homogeneous factorisation of order $p^l$ and
$\F_1'$ is a  circulant homogeneous factorisation of order $p^{d-l}$. Then
$\F_1'$ can not be a lexicographic product of two smaller
 circulants homogeneous factorisations. By induction, $\F_1'$ is normal.

Suppose first that $l\ge n$.   By Corollary \ref{lexico}, for each $i\in \{1,\ldots,k\}$, $rZ_{p^l}\subseteq T_i$ for any $r\in
T_i^2$ with $o(r)> p^l$. Since $T_i^2=S_i^2 (1\le i\le k)$, we have
$sZ_{p^l}\subseteq S_i$ for any $s\in
S_i^2$ with $o(s)> p^l$. By Corollary \ref{lexico}, $\F= \F_1[\F_2]$ where
$\F_2$ is a   circulant homogeneous factorisation of order $p^l$ and
$\F_1$ is a  circulant homogeneous factorisation of order $p^{d-l}$.

Suppose next that $l<n$. We will show that $\F$ is  normal in this case.

 Consider first the induced   sub-factorisation $\mathcal A'$ of $\F'$ on the block $Z_{p^n}$.
 Let $\t=\s|_{Z_{p^n}}\in \Aut(Z_{p^n})$. Then it follows from Lemma \ref{block} that $\t: T_i^1\to T_{i+1}^1$ is a cyclic isomorphism of $\mathcal A'$ and so $\A'$ is normal. Since $\s$ satisfies Lemma \ref{orders} and $o(\t)|o(\s)$,    the cyclic isomorphism $\t$  of $\A'$ satisfies Lemma \ref{orders}, that is   each prime divisor   of the order $o(\t)$ divides $k$.


 On the other hand
$\F'=\F'_1[\F'_2]$, and recall that  for each $i\in \{1,\ldots,k\}$, $rZ_{p^l}\subseteq T_i$ for any $r\in
T_i$ with $o(r)> p^l$. By Remark \ref{lexire}  we may suppose the factor digraphs of $\F'_1$ are the quotient Cayley digraphs $\ov\si_i =\Cay(Z_{p^d}/Z_{p^{l}},\ov  {T_i})$ and $\ov {T_i}=\{\ov r=rZ_{p^{l}}\mid r\in T_i, o(r)>p^l\}$ where $1\le i\le k$. And the induced natural cyclic isomorphism $\bar \s:\ov T_i\to \ov T_{i+1}$.




By induction, we have  seen that
$\F_1'$ is normal, and so by Lemma \ref{normalcyc} there exists a cyclic isomorphism $\bar\psi\in \Aut(Z_{p^d}/Z_{p^{l}})$
 satisfying Lemma \ref{orders} such that  $\bar \psi: \ov T_i\to \ov T_{i+1}$.  By Lemma \ref{auto}, we may also assume that  $\bar\psi$ is induced by
$\psi\in \Aut(Z_{p^d})$ where $o(\psi)=o(\bar\psi)$. In particular $\psi: T_i^2\to T^2_{i+1},  (1\le i
\le k,$  reading the subscript modulo $k$).

  Since  $l<n$, we have $rZ_{p^l}\subseteq T_i^1$ for any $r\in
T_i^1$ with $o(r)> p^l$, and so the  normal circulant homogeneous factorisation
$\mathcal A'=\mathcal A'_1[\mathcal A'_2]$ is also of lexicographic product type where $\A'_1$ is of  order $p^{n-l}$ and $\A_2'$ is of order $p^l$. Again by Remark \ref{lexire},
$\A'_1$ is the quotient factorisation  induced  by  $\A'$ on the block system $\{xZ_{p^l}|x\in Z_{p^n}\}$, and so the induced automorphism $\bar \t\in \Aut(Z_{p^n}/Z_{p^l})$ is a cyclic isomorphism of $\A'_1$. As $\t$ satisfies Lemma \ref{orders}, by Lemma \ref{auto}, $o(\bar\t)=o(\t)$. On the other hand, since $Z_{p^n}/Z_{p^{l}}< Z_{p^d}/Z_{p^{l}}$,
 we can also view $\A'_1$ as the induced  sub-factorisation by $\F_1'$ on the block $Z_{p^n}/Z_{p^{l}}$, and so $\bar \psi|_{Z_{p^n}/Z_{p^{l}}}$ is also a cyclic isomorphism of $\A'_1.$ This forces  $o(\psi|_{Z_{p^n}})=o(\bar\psi|_{Z_{p^n}/Z_{p^{l}}})=o(\bar \t)=o(\t)$
 by Lemma \ref{auto} and Lemma \ref{use}. Next we apply Lemma \ref{use}   to $\A'$,  then $\psi|_{Z_{p^n}}$ is also a cyclic isomorphism of $\A'$ and hence $\psi|_{Z_{p^n}}$ induces a transitive action on $\{T_i^1| i=1,\ldots,k\}$. Note that $\bar \psi: \ov T_i\to \ov T_{i+1} (1\le i\le k)$ where $\ov {T_i}=\{\ov r=rZ_{p^{l}}\mid r\in T_i, o(r)>p^l\}$  and $l<n$. We deduce that $\psi|_{Z_{p^n}}: T_i^1\to T_{i+1}^1$.
  Let $\A$ be the induced sub-factorisation   on $Z_{p^n}$ by $\F$.  Applying Lemma \ref{use} again to $\A$, we have $\psi|_{Z_{p^n}}$ is also a cyclic isomorphism of $\A$ and so  induces a transitive action on $\{S_i^1| i=1,\ldots,k\}$. By Lemma \ref{induction}, $T_i^0=S_i^0$. and so it is easy to deduce that  $\psi|_{Z_{p^n}}: S_i^1\to S_{i+1}^1$. It then follows from $S_i^2=T_i^2$ that $\psi: S_i\to S_{i+1} $ and so $\psi\in \Aut(Z_{p^d})$ is a cyclic isomorphism of $\F$. Therefore $\F$ is normal. This completes the proof.\qed









\begin{thebibliography}{99}


\bibitem{GLPP}
M. Giudici, C. H. Li, P. Potocnik and C. E. Praeger, Homogeneous factorisations of graph products, \emph{Discrete Math.} 308 (2008) 3652--3667.
\bibitem{godsil}
C. D. Godsil, On the full automorphism group of a graph,
\emph{Combinatorica} 1 (1981), 243--256.



\bibitem{kovacs}
I. Kov\'acs, Classifying arc-transitive circulants, \emph{J.
Algebraic Combin.} 20 (2004), 353--358.


\bibitem{l6}
C. H. Li, Finite abelian groups with the m-DCI property,
{\it Ars Combin.} 51(1999), 77--88.



\bibitem{li-arc-circ}
C. H. Li, Permutation groups with a cyclic regular subgroup and arc
transitive circulants, \emph{J. Algebraic Combin.} 21 (2005),
131--136.




\bibitem{cons-sc}
C. H. Li and C. E. Praeger, Constructing homogeneous factorisations
of complete graphs and digraphs, {\it Graphs Combin.} {\bf 18}
(2002), no. 4, 757--761.
\bibitem{lipraeger1}
 C.H.Li and C.E.Praeger, On partitioning the orbitals of a transitive permutation group, \emph{Trans. Amer. Math. Soc.} 355 (2003) 637--657.
\bibitem{lipraeger}
C. H. Li and C. E. Praeger,
On finite permutation groups with a transitive cyclic subgroup,
{\it J. Algebra} 349, (2012) 117--127.

\bibitem{lsx}
C. H. Li, S. Sun and J. Xu, Self-complementary circulants of prime-power order, \emph{SIAM J. Discrete
Math.} 28 (2014), no. 1, 8--17.

\bibitem{pls}
C.E. Praeger, C.H.Li and L. Stringer, Common circulant cyclic homogeneous factorisations of complete graphs, \emph{Discrete Math.} 309 (2009), 3006--3012.

\bibitem{Xu-survey}
M. Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs,
{\it Discrete Math.} {\bf 182} (1998), 309--320.



\end{thebibliography}

\end{document}

 