Circulant homogeneous factorisations of complete digraphs K p d with p an odd prime

Let F = (Kn,P) be a circulant homogeneous factorisation of index k, that means P is a partition of the arc set of the complete digraph Kn into k circulant factor digraphs such that there exists σ ∈ Sn permuting the factor circulants transitively amongst themselves. Suppose further such an element σ normalises the cyclic regular automorphism group of these circulant factor digraphs, we say F is normal. Let F = (Kpd ,P) be a circulant homogeneous factorisation of index k where pd, (d > 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.


Introduction
Let Γ = (V, A) be a digraph with vertex set V = V (Γ) and arc set A = A(Γ).A factorisation of Γ is a partition P = {P 1 , . . ., P k } of A, and is denoted by the pair (Γ, P).This gives rise to factor digraphs, Γ i = (V, P i ) for i = 1, . . ., k. Moreover the integer k = |P| is called the index of the factorisation and |V | is called the order of the factorisation.
An automorphism of a factorisation (Γ, P) is an automorphism of the digraph Γ that preserves the partition P. The automorphism group Aut(Γ, P) consists of all automorphisms of (Γ, P).A factorisation (Γ, P) is called transitive if Aut(Γ, P) induces a transitive action on P; further (Γ, P) is called homogeneous if it is transitive and in addition the kernel of Aut(Γ, P) acting on P is transitive on the vertex set V (Γ).That is there exists a subgroup G Aut(Γ, P) that permutes the parts P i transitively, and the kernel )) of G acting on P is vertex transitive on each of the Γ i .Hence G induces isomorphisms between each pair of the factor digraphs.To emphasise the groups M and G, we say that the factorisation (Γ, P) is (M, G)-homogeneous.Such a factorisation is also denoted by (Γ, P, M, G).We say a homogeneous factorisation (Γ, P, M, G) is cyclic if the induced group G P is cyclic.
A digraph is called a circulant if it has a cyclic group of automorphisms which is regular on the vertex set.(A permutation group is regular if it is transitive and the only element that fixes a point is the identity.)Let (Γ, P) be (M, G)-homogeneous, suppose further that M contains a regular cyclic subgroup.Then all of the factor digraphs Γ i are circulants relative to the same cyclic regular subgroup, and (Γ, P) is called a 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 Γ, Γ, where Γ is a vertex transitive digraph isomorphic to its complement Γ, that is, Γ is a vertex transitive self-complementary directed graph.Suppose further that the factors Γ, Γ are undirected.Then Γ is a vertex transitive self-complementary undirected graph.Moreover it is easy to see that self-complementary circulants (Γ, Γ) 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 [7,8].In [9], 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 [9].
Let F = (K n , P) be a circulant homogeneous factorisation.An automorphism τ (∈ Aut(F)) is called a cyclic isomorphism of F if τ is transitive on P. Suppose further that F is (M, G)-homogeneous.It is proved in [8,Theorem 4.1] that F must be cyclic, that is there exists σ ∈ G \ M such that G = M, σ and G P = σ P .Such an element σ is a cyclic isomorphism of F, and we often refer to (K n , P, M, σ) as this factorisation when we wish to emphasise this cyclic isomorphism.Moreover, let Z n ( 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 σ 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 normal.In this case, σ can be viewed as an automorphism of the cyclic group Z n , and we can construct such factorisations easily, see Construction 4.3 for more details.We also give examples of non-normal circulant homogeneous factorisations in Proposition 4.8.
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 [1,6].
For two digraphs the electronic journal of combinatorics 24(2) (2017), #P2.27 For i = 1, 2, let F i = (K n i , P i , M i , σ i ) be a circulant homogeneous factorisation of index k where σ i is a cyclic isomorphism of F i .Let P i = {P i,1 , . . ., P i,k } and suppose that σ i : P i,j → P i,j+1 (reading the second subscript modulo k).Write the vertex set In this paper, we will classify circulant homogeneous factorisations of a complete digraph K p d where p is an odd prime and d 1.The main theorem is the following.

Preliminary results
From now on, we always assume that p d (d 1) is an odd prime power.
A finite permutation group is called a c-group if it contains a cyclic regular subgroup.A precise list of primitive c-groups is given in the following lemma.
Corollary 2.2.Suppose that X is a solvable primitive c-group on Ω where |Ω| = p d is an odd prime power.Then |Ω| = p and Z p X AGL(1, p).
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 Z n -circulant is a Cayley digraph Γ = Cay(Z n , S) with vertex set Z n and arc set A(Γ) = {(g, sg) | g ∈ Z n , s ∈ S}, for some nonempty subset S of Z n \ {1}.We also denote by Ẑn the right regular representation of the group Z n .Then each Z n -circulant Γ admits Ẑn 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 Ẑn in Sym(Z n ), and Aut( In fact the normaliser N Aut(Γ) ( Ẑn ) = Ẑn Aut(Z n , S), see for example [2,11].The Cayley digraph Γ = Cay(Z n , S) is said to be a normal circulant if Ẑn is normal in Aut(Γ), or equivalently, if Aut(Γ) = Ẑn Aut(Z n , S).
A circulant is called arc-transitive if its automorphism group is transitive on the arc set.The finite arc-transitive circulants were classified independently by István Kovács [3] and Cai Heng Li [5] in 2004.The following result concerning arc-transitive circulants of order p d is an immediate corollary of Theorem 1.3 in [5], (just note that in [5,Theorem 1.3], the orders of the deleted lexicographic product type digraphs cannot be a prime power).
Theorem 2.3.Let Γ = Cay(Z p d , S) be a connected arc transitive directed circulant of order p d where p is an odd prime and d 1 is an integer.Then one of the following holds: (iii) There exists an arc-transitive circulant Σ of order p d−i such that Γ = Σ[K p i ] where 1 i < d.Let Z p i Z p d be the subgroup of order p i .Then sZ p i ⊆ S for any s ∈ S.
Let Ω be the vertex set of K n .Then a factorisation (K n , P) is also simply denoted by (Ω, P).
Let F = (K n , P) be a (M, G)-homogeneous factorisation of index k where P = {P 1 , . . ., P k }, and assume that G is imprimitive on the vertex set of K n .Let B be a block of G. Let P B i = P i ∩ (B × B) and P B = {P B 1 , P B 2 , . . ., P B k }.Then the factorisation (B, P B ) is called the induced sub-factorisation of F on the block B. Let (K n , P) be a circulant (M, G)-homogeneous factorisation of index k where P = {P 1 , . . ., P k } and identify the vertex set with the regular group Z n .Choose the point 1 ∈ Z n , let P i (1) = {α ∈ Z n |(1, α) ∈ P i } and let P(1) = {P 1 (1), . . ., P k (1)}.Then P(1) is a partition of Z n \ {1}, and the factor digraphs Γ 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 [7,Lemma 2.3] easily.
2. Let P(1) = {P 1 (1), . . ., P k (1)} be a partition of Z n \{1}.Define Γ i = Cay(Z n , P i (1)) and let P i be the set of arcs of Γ i .Then P = {P 1 , . . ., P k } is a partition of the arc set of K n .Let G be such that Ẑn G Sym(Z n ).Suppose the point stabilizer G 1 leaves P(1) invariant, and acts transitively on P(1).Then G = Ẑn G 1 leaves P invariant, and acts transitively on P. Let M be the kernel of the action of G on P. Then (K n , P) is a circulant (M, G)-homogeneous factorisation.
In the papers [8,10], circulant homogeneous factorisations of complete digraphs K n have been studied.The following theorem gives some basic properties of such factorisations.
Theorem 2.6.([8, Theorem 4.1]) Let (K n , P) be a circulant (M, G)-homogeneous factorisation of index k.Then the following statements hold: (i) G is soluble and (ii) for each prime divisor r of n, k|(r − 1).
We list some results concerning the induced sub-factorisations in the following lemma, the proofs of these results can be found in [8,9].Lemma 2.7.Let (K n , P) be a circulant (M, G)-homogeneous factorisation of index k where P = {P 1 , . . ., P k } and identify the vertex set of K n with the regular group Z n .
(1) Let ∆ be a block of G with order m where m|n.Then ∆ = xZ m is a coset of the subgroup Z m , and the induced sub-factorisation (∆, (2) Let σ ∈ G be a cyclic isomorphism which fixes the point 1 and maps P i → P i+1 (reading the subscript modulo k).Let ∆ = Z m be a block of G. Then σ fixes Z m setwise and σ| Zm is a cyclic isomorphism of the induced sub-factorisation (∆, P ∆ ) where Then G 1 maps elements of order p i to elements of order p i for i = 1, . . ., d. (2) Since σ fixes 1 and 1 ∈ ∆, σ fixes the block ∆.Since Therefore σ| Zm is a cyclic isomorphism of the induced sub-factorisation (∆, P ∆ ). ( and so maps elements of order p i to elements of order p i .(See also [9,Lemma 4.3,Corollary 4.4].) We also need the following useful lemma.Since the circulant homogeneous factorisation must be cyclic by Theorem 2.6 (1), the following lemma is a direct corollary of [7,Lemma 5.2].
the electronic journal of combinatorics 24(2) (2017), #P2.27Lemma 2.8.([7, Lemma 5.2]) 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 B be a nontrivial G-invariant partition of V (K n ).Then each element of G \ K fixes exactly one block of B.
We finish this section by giving the following easy proposition.Proposition 2.9.Suppose that F = (K p , P) is a circulant homogeneous factorisation of order p where p is an odd prime.Then F is normal.
Proof.Let G = Aut(F) and let M be the kernel of G acting on P. By Theorem 2.6 (1) G is solvable, and hence G is a primitive solvable c-group.By Corollary 2.2, Ẑp G Ẑp Z p−1 .Any element σ ∈ G \ M is a cyclic isomorphism of F and hence F is normal by definition.
3 Lexicographic product constructions Suppose (K n , P, X, Y ) is a circulant homogeneous factorisation of index k and let σ ∈ Y be a cyclic isomorphism.Recall that we also refer to (K n , P, X, σ) as this factorisation.
Moreover, for any j ∈ {1, . . ., k}, let Cay(Z n 1 n 2 , S j ) be the factor digraphs of Proof.Write the vertex set K n 1 n 2 as V (K n 1 ) × V (K n 2 ).Then it is easy to see that P = {P 1,j [P 2,j ]|1 j k} is a partition of the arc set of K n 1 n 2 .
Let X and σ be the induced permutation groups of X and σ on the block system X. Next we define an induced quotient factorisation F 1 on this block system Z n /Z n 2 .For j = 1, . . ., k, let P 1,j (1) = P j (1) \ Z n 2 and P 2,j (1) = P j (1) ∩ Z n 2 .
Corollary 3.3.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 ∈ {1, . . ., 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 where F 2 is of order p m .
2. For any j ∈ {1, . . ., k} and any s ∈ S j with o(s) > p m , sZ p m ⊆ S j .
Proof.Just note that by Lemma 2.7 (4), Z p m is a block of Aut(F), the result follows from Lemma 3.1 and 3.2.
Moreover we have the following useful remark.
Remark 3.4.Let F = (K n , P, X, σ) be a circulant homogeneous factorisation of index k where σ is a cyclic isomorphism fixing point 1 and let P = {P 1 , . . ., 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 X, σ .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 ∈ Z n } defined as in the proof of Lemma 3.2.In particular, with the notation in Lemma 3.2, the factor digraphs of F 1 are the quotient cayley digraphs Γ 1,j = Cay(Z n /Z n 2 , P 1,j (1)) and }. Suppose that σ : P j → P j+1 .Then σ : Γ 1,j → Γ 1,j+1 .
the electronic journal of combinatorics 24(2) (2017), #P2.27 4 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.
Lemma 4.1.Let F = (K p d , P) be a circulant (X, Y )-homogeneous factorisation of index k and let τ be a cyclic isomorphism.Then k|(p − 1) and k|o(τ ).Moreover there exists a cyclic isomorphism σ ∈ Y such that σ fixes 1 and r|k for each prime divisor r of the order o(σ).
Proof.By Theorem 2.6, k|(p − 1).Since τ is transitive on P and |P| = k, k|o(τ ).By Theorem 2.6 again, there exists a cyclic isomorphism σ ∈ Y \ X such that Y = X, σ and Y P = σ P is a cyclic group of order k.Since Ẑp d X is vertex transitive, we may assume that σ fixes the point 1.Let q be a prime such that (q, k) = 1.For any positive integer m, σ q m (∈ Y ) is still transitive on P and so is also a cyclic isomorphism of F. Therefore replacing σ by some power of σ if necessary, we may assume r|k for each prime divisor r of the order o(σ).
Remark 4.2.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 σ of F, σ fixes the point 1.Moreover if r|k for each prime divisor r of the order o(σ), then we say the cyclic isomorphism σ satisfies Lemma 4.1.
the electronic journal of combinatorics 24(2) (2017), #P2.27By Lemma 2.5 (2) (taking G as Z p d , σ ), F = (K p d , P) is a normal circulant homogeneous factorisation of index k and σ is a cyclic isomorphism of this factorisation.Denote J = {j 1 , . . ., j t }.Since this construction F depends on the choice of σ and J, we also denote this construction by F σ,J .
Note that there do exist p, d, k, σ satisfying the conditions of Construction 4.3.Conversely, suppose that (K p d , P) is a normal circulant homogeneous factorisation of index k.Then each factor digraph Γ i is a circulant Cay(Z p d , P i (1)) where P i (1) = {α ∈ Z n |(1, α) ∈ P i }.Let σ ∈ Aut(Z p d ) be a cyclic isomorphism.As proved in Lemma 4.1, we may assume that each prime divisor of the order o(σ) divides k.Therefore σ induces a transitive action on P(1) = {P 1 (1), . . ., P k (1)} and σ k fixes each P i (1) for i = 1, . . ., k.As defined above in Construction 4.3, let ∆ 1 , . . ., ∆ t be the orbits of σ on Z p d \ {1}.And for each i, σ k divides ∆ i into k orbits, say ∆ i,1 , ∆ i,2 , . . ., ∆ i,k such that ∆ σ i,j = ∆ i,j+1 .It is easy to show that there exists j 1 , . . ., j t ∈ {1, . . ., k} such that jt } and P i (1) = P 1 (1) σ i−1 as in Construction 4.3.Therefore Construction 4.3 provides us with a method for constructing all normal circulant homogeneous factorisations of order p d , we write this result in the following proposition.Proposition 4.4.Let F be a normal circulant homogeneous factorisations of order p d and index k.And let σ ∈ Aut(Z p d ) be a cyclic isomorphism of F which satisfies Lemma 4.1.With above notation, there exists J = {j 1 , . . ., j t } such that F = F σ,J which is defined in Construction 4.3.
We will need the following easy lemma for the proof of the main theorem.
Conversely, suppose that τ is a cyclic isomorphism.Then τ is transitive on P. Let H = σ, τ .Then H is transitive on P. Note that σ, τ ∈ Aut(Z Lemma 4.7.Suppose that F = (K p d , P) is a normal circulant homogeneous factorisation of index k where P = {P 1 , .., P k }.Let σ be a cyclic isomorphism (not required to be a group automorphism) such that σ : P i → P i+1 (reading the subscript modulo k).Then there exists a cyclic isomorphism τ ∈ Aut(Z p d ) satisfying Lemma 4.1 and τ : P i → P i+1 too.
Proof.Since F is normal, by definition there exists a cyclic isomorphism τ ∈ Aut(Z p d ) satisfying Lemma 4.1.Consider Y = Ẑp d , σ, τ and let X be the kernel of Y acting on P. Let σ = σ P and τ = τ P .Then Y /X = σ = τ , and hence σ = τ l for some positive integer.Replacing τ by τ l , we have that τ : P i → P i+1 as required.
Lastly in this section we give examples of non-normal circulant homogeneous factorisations of order p 2 .Note that there do exist p, k, σ 1 , σ 2 satisfying the conditions of the following proposition.Proposition 4.8.Let p be an odd prime and k( 2) a positive integer such that k|(p − 1).Suppose further that σ 1 , σ 2 ∈ Aut(Z p ) such that o(σ 1 ) = k and o(σ 2 ) = mk where m 2 and each prime divisor of m divides k.By Construction 4.3, there exist circulant homogeneous factorisations F i of index k and order p such that σ i is a cyclic isomorphism of F i respectively.Take Then F is a non-normal circulant homogenous factorisation of order p 2 .Proof.By Lemma 3.1 F is a circulant homogeneous factorisation of order p 2 .Suppose conversely that F is normal and σ ∈ Aut(Z p 2 ) is a cyclic isomorphism of F satisfying Lemma 4.1.By Remark 3.4, F 2 is the sub-factorisation induced on the block Z p .By Lemma 2.7 (2), σ| Zp ∈ Aut(Z p ) is also a cyclic isomorphism of F 2 .By Lemma 4.6, o(σ| Zp ) = o(σ 2 ).It follows from Lemma 4.5 that o(σ 2 ) = o(σ).On the other hand, F 1 is the quotient factorisation of the block system {gZ p |g ∈ Z p 2 } by Remark 3.4, and the induced σ ∈ Aut(Z p 2 /Z p ) is also a cyclic isomorphism of F 1 .Still by Lemma 4.6 and Lemma 4.5, we have o(σ 1 ) = o(σ) = o(σ) which contradicts the fact that o(σ 1 ) = o(σ 2 ).Therefore F is not normal.

Proof of Theorem 1.1
We first study the structures of circulant homogeneous factorisations of order p d .Lemma 5.1.Let (K p d , P, X, Y ) be a circulant homogeneous factorisation of index k, and let σ ∈ Y be a cyclic isomorphism.Let Γ i = Cay(Z p d , S i ) be the factor digraphs where the electronic journal of combinatorics 24(2) (2017), #P2.27Let X 1 and Y 1 be the point stabilizers of 1 in X and Y respectively.Then Y 1 = X 1 , σ and suppose that σ : S i → S i+1 (1 i k), reading the subscript modulo k.
First take any g ∈ S 1 with o(g) In particular, Γ g is not the complete graph K p d .Applying Theorem 2.3, we have either In the latter case, we claim that and so gZ p is fixed by xσ i−1 .Since gZ p = Z p , xσ i−1 fixes at least two blocks gZ p and Z p , contradicting Lemma 2.8.Hence gZ p ⊂ S 1g .
For any g i ∈ S i (i 2) with o(g i ) = p d , we consider the orbital digraph Γ g i of Y with arc set ∆ = (1, g i ) Y as well.Applying the same argument as above repeatedly, it is easy to deduce that either Ẑp d Y and σ ∈ Aut(Z p d ), or for any i ∈ {1, . . ., k}, sZ p ⊂ S i where s ∈ S i with o(s) = p d .Proposition 5.2.Let (K p d , P, X, Y ) be a circulant homogeneous factorisation of index k, and let σ ∈ Y be a cyclic isomorphism.Let Γ i = Cay(Z p d , S i ) be the factor digraphs.Then there exists n ∈ {1, . . ., d} such that Ẑp n Y Z p n Z p n and σ| Z p n ∈ Aut(Z p n ).Moreover, for each i ∈ {1, . . ., k} and each s ∈ S i with o(s) > p n , we have sZ p ⊂ S i .
Proof.If d = 1, then take n = 1, the result follows from Corollary 2.2.Suppose next that d 2. By Lemma 5.1, we may assume that for any i ∈ {1, . . ., k}, sZ p ⊂ S i where s ∈ S i with o(s) = p d .By Lemma 2.7 (4), the subgroup Z p d−1 is a block of Y and the induced factorisation Z p n and for each i ∈ {1, . . ., k} and each s ∈ S i with o(s) > p n , sZ p ⊂ S i .Thus for any i ∈ {1, . . ., k}, we divide the Cayley subset S i into the following two parts.Let Then Lemma 5.3.Suppose that F = (K p d , P, X, Y ) is a circulant homogeneous factorisation of index k with d 2, and let σ ∈ Y be a cyclic isomorphism.For any i ∈ {1, . . ., k}, let Γ i = Cay(Z p d , S i ) be the factor digraphs, and suppose that σ : Γ i → Γ i+1 (reading the subscript modulo k).Suppose further that n, S 1 i , S 2 i are defined as above in (1).Then there exists a factorisation F = (K p d , P ) satisfying the following conditions.
(i) F is also a circulant (X, Y )-homogeneous factorisation of index k.Let Σ i = Cay(Z p d , T i ) be the corresponding factor digraphs of F .Then σ : Σ i → Σ i+1 is a cyclic isomorphism of F .
(ii) For any i ∈ {1, . . ., k} and any r ∈ T i such that o(r) > p, we have rZ p ⊆ T i .(This implies F is of lexicographic product form by Corollary 3.3.)(iii) For any i ∈ {1, . . ., k}, let (iv) For any i ∈ {1, . . ., k}, let and let τ = σ| Z p n .Then Y n = X n , τ .By Lemma 2.7 the induced sub-factorisation (on the block forms a complete block system of Y n on Z p n .As τ normalises L, (gZ p ) Lτ = (gZ p ) τ L .For any gZ p ∈ B 1 such that gZ p = Z p , it follows from Lemma 2.8 that g τ i Z p = (gZ p ) x for any x ∈ L and 1 i < k.Thus we may suppose that L has km orbits (for some integer m 1) on B 1 \ {Z p } and after relabelling if necessary, we may suppose that ∆ j,1 , ∆ j,2 , . . ., ∆ j,k are L-orbits on B 1 \ {Z p } such that ∆ τ j,i = ∆ j,i+1 where j ∈ {1, . . ., m}, i ∈ {1, . . ., k}. Let . By Lemma 2.7 (4) and our construction it is easy to see that the vertex stabilizer the electronic journal of combinatorics 24(2) (2017), #P2.27X 1 of X fixes each T i (i = 1, . . ., k) and σ : T i → T i+1 .Define Σ i = Cay(Z p d , T i ), (i = 1, . . ., k) and let P i be the set of arcs of Σ i and let P = {P 1 , . . ., P k }.By Lemma 2.5 (2) F = (K p d , P ) is a circulant (X, Y )-homogeneous factorisation of index k, and σ is a cyclic isomorphism of F .The results (ii), (iii) and (iv) follow from the construction of F obviously.
We are ready to prove Theorem 1.1.Proof of Theorem 1.1: We proceed by induction on d.If d = 1, the result follows from Proposition 2.9.Assume inductively the result holds for circulant homogeneous factorisations of order p d−1 where d 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 σ be a cyclic isomorphism satisfying Lemma 4.1 and let Γ i = Cay(Z p d , S i ) be the factor digraphs (1 i k).We assume that σ : Γ i → Γ i+1 (reading the subscript modulo k).By Proposition 5. .By Lemma 5.3, there exists a circulant (X, Y ) homogeneous factorisation F = (K p d , P ) such that σ is also a cyclic isomorphism of F .Let Σ i = Cay(Z p d , T i ) be the corresponding factor digraphs of F .Then σ : Σ i → Σ i+1 , (reading the subscript modulo k).Setting 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 n.By Corollary 3.3, for each i ∈ {1, . . ., k}, rZ p l ⊆ T i for any r ∈ T 2 i with o(r) > p l .Since T 2 i = S 2 i (1 i k), we have sZ p l ⊆ S i for any s ∈ S 2 i with o(s) > p l .By Corollary 3.3, 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 A of F on the block Z p n .Let τ = σ| Z p n ∈ Aut(Z p n ).Then it follows from Lemma 2.7 that τ : T 1 i → T 1 i+1 is a cyclic isomorphism of A and so A is normal.Since σ satisfies Lemma 4.1 and o(τ )|o(σ), the cyclic isomorphism τ of A satisfies Lemma 4.1, that is each prime divisor of the order o(τ ) divides k.
On the other hand F = F 1 [F 2 ], and recall that for each i ∈ {1, . . ., k}, rZ p l ⊆ T i for any r ∈ T i with o(r) > p l .By Remark 3.4 we may suppose the factor digraphs of F 1 are the quotient Cayley digraphs Σ i = Cay(Z p d /Z p l , T i ) and T i = {r = rZ p l | r ∈ T i , o(r) > p l } where 1 i k.And the induced natural cyclic isomorphism σ : T i → T i+1 .
By induction, we have seen that F 1 is normal, and so by Lemma 4.7 there exists a cyclic isomorphism ψ ∈ Aut(Z p d /Z p l ) satisfying Lemma 4.1 such that ψ : T i → T i+1 .By the electronic journal of combinatorics 24(2) (2017), #P2.27Lemma 4.5, we may also assume that ψ is induced by ψ ∈ Aut(Z p d ) where o(ψ) = o( ψ).In particular ψ : T 2 i → T 2 i+1 , (1 i k, reading the subscript modulo k).Since l < n, we have rZ p l ⊆ T 1 i for any r ∈ T 1 i with o(r) > p l , and so the normal circulant homogeneous factorisation A = A 1 [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 3.4, A 1 is the quotient factorisation induced by A on the block system {xZ p l |x ∈ Z p n }, and so the induced automorphism τ ∈ Aut(Z p n /Z p l ) is a cyclic isomorphism of A 1 .As τ satisfies Lemma 4.1, by Lemma 4.5, o(τ ) = o(τ ).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 ψ| Z p n /Z p l is also a cyclic isomorphism of A 1 .This forces o(ψ| Z p n ) = o( ψ| Z p n /Z p l ) = o(τ ) = o(τ ) by Lemma 4.5 and Lemma 4.6.Next we apply Lemma 4.6 to A , then ψ| Z p n is also a cyclic isomorphism of A and hence ψ| Z p n induces a transitive action on {T 1 i |i = 1, . . ., k}.Note that ψ : T i → T i+1 (1 i k) where T i = {r = rZ p l | r ∈ T i , o(r) > p l } and l < n.We deduce that ψ| Z p n : T 1 i → T 1 i+1 .Let A be the induced sub-factorisation on Z p n by F. Applying Lemma 4.6 again to A, we have ψ| Z p n is also a cyclic isomorphism of A and so induces a transitive action on {S 1 i |i = 1, . . ., k}.By Lemma 5.3, T 0 i = S 0 i .and so it is easy to deduce that ψ| Z p n : S 1 i → S 1 i+1 .It then follows from S 2 i = T 2 i that ψ : S i → S i+1 and so ψ ∈ Aut(Z p d ) is a cyclic isomorphism of F. Therefore F is normal.This completes the proof.

Theorem 1 . 1 .
Let F = (K p d , P) be a circulant homogeneous factorisation of index k where p d (d 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 1).

Lemma 2 . 4 .
([7, Lemma 4.1]) 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 P B B .

Proof. ( 1 )
Let B be a block of G such that 1 ∈ 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 ∆ is a block of order m, and x ∈ ∆.Then ∆ = xZ m as required.Hence M ∆ ∆ contains the regular subgroup Z m .It follows from Lemma 2.4 that the induced sub-factorisation (∆, P ∆ ) is a circulant (M ∆ ∆ , G ∆ ∆ )-homogeneous factorisation of index k.

Lemma 4 . 5 .
Let p d be an odd prime power such that d 2, and let σ ∈ Aut(Z p d ) such that o(σ)|(p − 1).Let Z p i Z p d be a subgroup.Then (1) o(σ| Z p i ) = o(σ).(2) Let σ ∈ Aut(Z p d /Z p i ) be the automorphism of the quotient group induced by σ.Then o(σ) = o(σ).

Lemma 4 . 6 .
Suppose that (K p d , P) is a normal circulant homogeneous factorisation of index k and σ ∈ Aut(Z p d ) is a cyclic isomorphism satisfying Lemma 4.1.Let τ ∈ Aut(Z p d ) such that k|o(τ ) and each prime divisor of o(τ ) divides k.Then τ is a cyclic isomorphism of (K p d , P) if and only if o(σ) = o(τ ).Proof.By assumption we may assume that o(σ) = mk, o(τ ) = nk where m, n are positive integers and each prime divisor of m and n divides k.

1 i k .
Then either Ẑp d Y and σ ∈ Aut(Z p d ), or for any i ∈ {1, . . ., k} and any s ∈ S i with o(s) = p d , sZ p ⊂ S i .Proof.If d = 1 then Y is a soluble primitive c-group.It follows from Corollary 2.2 that Ẑp Y AGL(1, p) and σ ∈ Aut(Z p ). Next suppose that d 2.
1 and σ| Z p d−1 ∈ Aut(Z p d−1 ), or for any i ∈ {1, . . ., k}, sZ p ⊂ S i where s ∈ S i with o(s) = p d−1 .Continuing in this fashion (note that Ẑp Y Z p n , σ| Z p n ∈ Aut(Z p n ) and for any i ∈ {1, . . ., k}, sZ p ⊂ S i where s ∈ S i with o(s) > p n .Suppose (K p d , P, X, Y ) is a circulant homogeneous factorisation and denote the factor digraphs by Γ i = Cay(Z p d , S i ).By Proposition 5.2, there exists n ∈ {1, . . ., d} such that Ẑp n Y 2, there exists n ∈ {1, . . ., d} such that Ẑp n Y Z p n Z p n and σ| Z p n ∈ Aut(Z p n ).If n = d then F is normal as required.So we assume next that n < d.By Proposition 5.2, for each i ∈ {1, . . ., k} and each s ∈ S i with o(s) > p n , sZ p ⊂ S i .Let S 1 i = {s ∈ S i | o(s) p n } and S 2 i = {s ∈ S i | p d o(s) p n+1 }.Then S i = S 1 i ∪ S 2 i