Basic Tetravalent Oriented Graphs of Independent-Cycle Type

The family $\mathcal{OG}(4)$ consisting of graph-group pairs $(\Gamma, G)$, where $\Gamma$ is a finite, connected, 4-valent graph admitting a $G$-vertex-, and $G$-edge-transitive, but not $G$-arc-transitive action, has recently been examined using a normal quotient methodology. A subfamily of $\mathcal{OG}(4)$ has been identified as `basic', due to the fact that all members of $\mathcal{OG}(4)$ are normal covers of at least one basic pair. We provide an explicit classification of those basic pairs $(\Gamma, G)$ which have at least two independent cyclic $G$-normal quotients (these are $G$-normal quotients which are not extendable to a common cyclic normal quotient).


Introduction
Finite tetravalent graphs admitting a half-arc-transitive group action have been consistently and actively studied since the 1990s.Several different trends have emerged in this research, often focusing on a particular aspect of these objects, such as their alternating cycles [6,9,12], their normal quotients [1,2,14], their vertex stabilisers [5,7,11,21], or their role as medial graphs for regular maps on surfaces [8].In [2], a general framework was introduced for studying the family OG(4) of graph-group pairs (Γ, G), with Γ a finite connected tetravalent graph, and G a vertex-and edge-transitive, but not arc-transitive, group of automorphisms.This method (described in detail below) uses a normal quotient reduction, and has already been successfully used to study other families of graphs exhibiting particular symmetry conditions, see for instance [10,16,17].The basic aim of the method is to describe OG(4) using graph quotients arising from normal subgroups of the groups contained in the family.
Given a pair (Γ, G) ∈ OG(4), and a normal subgroup N of G, we may define a G-normal quotient graph Γ N of Γ.The vertices of Γ N are the N -orbits in V Γ, and there is an edge between two vertices of Γ N if and only if there is at least one edge between vertices from the corresponding N -orbits in V Γ.The group G then induces a group G N of automorphisms of Γ N , so that we obtain another graph-group pair (Γ N , G N ).The important result [2, Theorem 1.1] then tells us that taking a normal quotient of a pair (Γ, G) ∈ OG(4) either produces another pair (Γ N , G N ) ∈ OG(4) (and in this case Γ is a G-normal cover of Γ N ), or the quotient graph Γ N is isomorphic to one of K 1 , K 2 or C r for some r ≥ 3.In the latter case we say that the quotient is degenerate.
In light of this result, we say that a pair (Γ, G) ∈ OG (4) is basic if all of its G-normal quotients relative to non-trivial normal subgroups of G are degenerate.By [2,Theorem 1.1], it follows that every member of OG(4) is a normal cover of at least one basic pair.Hence the authors of [2] suggest that we may obtain a description of OG(4) by producing a good description of the basic pairs, and combining it with a theory to describe their G-normal covers.
With this aim in mind, the authors of [2] further divide the basic pairs in OG(4) into three types.A pair (Γ, G) ∈ OG( 4) is said to be basic of quasiprimitive type if the only degenerate G-normal quotient of Γ is K 1 .This occurs precisely when all non-trivial normal subgroups of G are transitive on V Γ; such a permutation group is called quasiprimitive.If the only degenerate G-normal quotients of a basic pair (Γ, G) ∈ OG(4) are the graphs K 1 or K 2 , and Γ has at least one G-normal quotient isomorphic to K 2 , then (Γ, G) is said to be basic of biquasiprimitive type.(The group G here is biquasiprimitive: it is not quasiprimitive but each nontrivial normal subgroup has at most two orbits.)The other basic pairs in OG(4) must have at least one normal quotient isomorphic to a cycle graph C r for some r ≥ 3, and these basic pairs are said to be of cycle type.
In the first paper on this topic [2], the authors manage to provide a general description of the basic pairs of quasiprimitive type [2,Theorem 1.3] by applying the structure theorem for finite quasiprimitive permutation groups given in [16].The basic pairs of biquasiprimitive type were analysed and described in [14], where a method analogous to the quasiprimitive case was used, this time applying the lessdetailed structure theorem for biquasiprimitive permutation groups in [18].A description of the basic biquasiprimitive pairs is given in [14,Theorem 1.1].
The remaining basic pairs are those having at least one cyclic normal quotient.As noted above, these pairs are said to be basic of cycle type.Since there is no general theory describing the groups appearing in these pairs, they are the most difficult to analyse and describe.These pairs are also more complex than both the quasiprimitive and biquasiprimitive basic pairs, in that there is much greater diversity in the possible degenerate quotients they can have.In particular, a basic pair (Γ, G) ∈ OG(4) of cycle type may have many non-isomorphic cyclic normal quotients.These quotients, despite all being cycles, may have different orders and may be G-oriented or G-unoriented depending on the G-action on the quotient graph, see [1,Theorem 1].
In trying to understand the possible types of degenerate quotients which may occur for basic pairs of cycle type, the importance of considering so-called 'independent' cyclic normal quotients was demonstrated in [1].Two cyclic normal quotients of a pair (Γ, G) ∈ OG(4) are said to be independent cyclic normal quotients if they are not extendable to a common cyclic normal quotient of (Γ, G).
If (Γ, G) is a basic pair of cycle type and does not have independent cyclic normal quotients, then all of its cyclic normal quotients will be either G-oriented, or all of its cyclic normal quotients will be G-unoriented (see [15,Section 3]).Hence the basic pairs of cycle type were further divided into three types in [15] as follows.A basic pair (Γ, G) ∈ OG(4) with cyclic normal quotients is 1. basic of oriented-cycle type if (Γ, G) does not have independent cyclic normal quotients and all of its cyclic normal quotients are G-oriented, 2. basic of unoriented-cycle type if (Γ, G) does not have independent cyclic normal quotients and all of its cyclic normal quotients are G-unoriented,

basic of independent-cycle type if (Γ, G) has independent cyclic normal quotients.
Through a careful analysis, a description of the basic pairs of oriented-cycle type and unoriented-cycle type was obtained in [15,Theorem 1.1].
On the other hand, the existence of independent cyclic quotients of a pair (Γ, G) ∈ OG(4) proves to be restrictive enough to allow for a description of all such pairs.In particular, by [1,Theorem 2], if a pair (Γ, G) has independent cyclic normal quotients then it is a G-normal cover of a pair (Γ, G) ∈ OG(4) which belongs to one of six infinite families appearing in [1, Table 1].The (underlying unoriented) graphs in these families are all either direct products of two cycles C r × C s for some r, s ≥ 3, or are induced subgraphs or standard double covers of these direct product graphs.
It follows that all basic pairs (Γ, G) of independent-cycle type belong to one of the families of examples in [1, Table 1].The objective of this paper is to identify which members of these families are in fact basic.Our main result is Theorem 1.It identifies explicitly these basic members.The graphs and groups mentioned in this theorem are all defined in detail in Definitions 1 and 2 in the next section.4) is basic of independent-cycle type if and only if it appears in one of the lines of Table 1.In each row p and q denote arbitrary odd primes.Theorem 1, when combined with [15, Theorem 1.1], provides a description of all the basic pairs of cycle type.We note in particular that our classification of the basic pairs of independent-cycle type is completely explicit.For all other types, apart from one explicit family of examples in [15, Theorem 1.1.1(a)],the analysis shows that, for a basic pair (Γ, G), the group G has a unique minimal normal subgroup N , and tight upper bounds are obtained on the number of simple direct factors of N , [2,13,15].
In the next section we set up our approach to proving Theorem 1 by outlining the necessary background theory.We then prove this theorem in two parts in Sections 3 and 4.

Preliminaries
All graphs in this paper are simple and finite.Given a graph Γ we will let V Γ and EΓ denote the sets of vertices and edges of Γ respectively.Given a vertex α ∈ V Γ we will let Γ(α) denote the set of neighbours of α.An arc of a graph Γ is an ordered pair of adjacent vertices.We will let AΓ denote the set of arcs of Γ.Given an arc (α, β) ∈ AΓ, we will call (β, α) its reverse arc.In particular, each edge {α, β} ∈ EΓ has two arcs associated with it, namely (α, β) and (β, α).For other basic graph-theoretic concepts, please refer to [4].

G-Oriented Graphs
For basic concepts of permutation group theory we refer the reader to [19].Given a graph Γ and a group G ≤ Aut(G), we say that Γ is G-vertex-transitive, G-edge-transitive, or G-arc-transitive if G is transitive on V Γ, EΓ, or AΓ respectively.
We will say that a graph Γ is G-oriented, with respect to a group G ≤ Aut(Γ), if G is transitive on V Γ and EΓ but is not transitive on AΓ.In the literature, such graphs are also called G-half-arctransitive, or are said to admit a half-transitive G-action.
Since G-oriented graphs are vertex-transitive they are necessarily regular and all of their connected components are isomorphic.Moreover, the valency of a G-oriented graph is necessarily even [20].It is easy to see that the connected 2-valent G-oriented graphs are oriented cycles.Hence the smallest nontrivial valency a G-oriented graph can have is 4.
We will let OG(4) denote the family of graph group pairs (Γ, G) where Γ is a connected 4-valent G-oriented graph.For a detailed overview of the family OG(4), and of G-oriented graphs in general, please see [2] or [15, Section 2].

Normal Quotients
Suppose that (Γ, G) ∈ OG(4) and let N be a nontrivial normal subgroup of G. Let B N be the partition of V Γ into the N -orbits, that is, B N = {(α N ) g : g ∈ G} where α is an arbitrary fixed vertex of Γ, and α N denotes the N -orbit containing α.In this case, B N is a G-invariant partition of V Γ, so we may define the G-normal-quotient graph of Γ with respect to N , denoted Γ N .The vertex set of Γ N is the set B N of N -orbits, and there is an edge between two vertices in Γ N if and only if there is at least one edge of Γ between vertices from the corresponding N -orbits.
The group G then induces a group G N of automorphisms of Γ N .Specifically, G N = G/K, where K is the kernel of the G-action on Γ N .By definition, N ≤ K and since K fixes all N -orbits setwise, it follows that the K-orbits are the same as the N -orbits, so Γ K = Γ N .However, K may be strictly larger than N .
It was shown in [2, Theorem 1.1] that for any (Γ, G) ∈ OG(4), and any nontrivial normal subgroup Following the authors of [2], we say that a pair As mentioned in the introduction, the basic pairs can be divided into three types (quasiprimitive, biquasiprimitive and cycle type) depending on their possible degenerate normal quotient graphs, and those of the first two types have been studied in [2] and [14] respectively.Here, we will only be concerned with the basic pairs of cycle type.
Recall that a pair (Γ, G) ∈ OG( 4) is basic of cycle type if it has at least one G-normal quotient which is isomorphic to a cycle graph C r for some r ≥ 3. Suppose that (Γ, G) is such a pair with cyclic normal quotient Γ N ∼ = C r , for some normal subgroup N of G and r ≥ 3.In such a case, we will always let N denote the largest normal subgroup of G (containing N ) which fixes each N -orbit setwise, that is, N will always refer to the kernel of the G-action on the set of N -orbits in V Γ.Since N ≤ N and these groups have the same orbits in V Γ, it will always be the case that Γ N = Γ N .
Since Γ N is a cyclic normal quotient of Γ, it follows from the fundamental analysis in [2, Theorem 1.1] that either Γ N is an oriented cycle and the induced group G := G/ N is its full automorphism group C r , or Γ N is an unoriented cycle and the induced group G := G/ N acts arc-transitively, and again is the full automorphism group D r , where D r denotes the dihedral group of order 2r.If the normal quotient (Γ N , G/ N ) of a pair (Γ, G) ∈ OG( 4) is isomorphic to (C r , C r ) for some r ≥ 3, then we will say that the quotient is G-oriented (or simply that it is oriented).If (Γ N , G/ N ) is isomorphic to (C r , D r ) then we will say that the quotient is G-unoriented (or simply that it is unoriented).
Of course, if Γ N is a G-oriented cyclic normal quotient, then the stabiliser G α of any vertex will fix the N -orbit containing α, and so G α ≤ N .On the other hand, if Γ N is G-unoriented then any vertex α will have one out-neighbour (and one in-neighbour) in each of the two adjacent N -orbits, hence any non-identity automorphism which fixes α must swap at least two N -orbits, in particular G α ∩ N = 1.
Two cyclic normal quotients Γ M and Γ N of (Γ, G) ∈ OG(4) are independent if the normal quotient 4) is basic of cycle type and Γ M and Γ N are independent cyclic normal quotients of Γ, then If a basic pair does not have independent cyclic normal quotients then all of its normal quotients must be oriented or all of its normal quotients must be unoriented (see the discussion in the introduction), and such basic pairs have been analysed in [15].The purpose of this paper is to identify those basic pairs (Γ, G) ∈ OG(4) of independent-cycle type, that is those basic pairs which have independent cyclic normal quotients.

Independent Cyclic Normal Quotients
A description of the pairs (Γ, G) ∈ OG(4) having independent cyclic quotients was provided in [1] via the following theorem.4), let α be a vertex, and suppose that (Γ, G) has independent cyclic normal quotients and the following hold: Before progressing further, we will give the definitions of the graphs and groups appearing in Table 2 (and also in Table 1 for Theorem 1) as they are stated in [1].Definition 1. [1, Definition 2.1] Let r, s be integers, each at least 3. Define the undirected graph Γ(r, s) to have vertex set X := Z r × Z s , such that a vertex (i, j) ∈ X is joined by an edge to each of the four vertices (i ± 1, j ± 1).Also, if r, s are both even define and let Γ + (r, s) = [X + ], the induced subgraph.Define the following permutations of X, for (i, j) ∈ X, µ : (i, j) and define the groups as in Table 3, where in lines 3 and 4 (r, s both even), we identify µ, ν, σ, τ with their restrictions to X + , and consider the subgroups G + (r, s) and H + (r, s) acting on X + .
Both even Also let (b) M t = µ t for t | r, and N t = ν t for t | s.If r and s are both even, we also consider the following subgroups restricted to their actions on X + : We will need to consider the standard double covers (sometimes called canonical double covers) of the graphs Γ(r, s) defined in Definition 1.The standard double cover of a graph Γ with vertex set X is the graph Γ 2 with vertex set and {x, y} is an edge of Γ.Note that Γ 2 has the same valency as Γ and twice the number of vertices.
We extend the automorphisms defined in Definition 1 to maps on X 2 as follows.
By Theorem 2, if a pair (Γ, G) ∈ OG(4) has independent cyclic normal quotients, then (Γ, G) is a normal cover of a pair (Γ, G) ∈ OG(4) appearing in Table 2, and so every basic pair of independentcycle type appears in Table 2, though of course not all such pairs are basic.We now make some preliminary remarks about our proof strategy.
Remark 1.Since every basic pair of independent-cycle type appears in Table 2, all we need to do in order to determine these basic pairs (and hence prove Theorem 1), is to decide which of the pairs in Table 2 are basic.For the purpose of doing this, we in fact only need to determine which of the pairs appearing in Rows 1, 2, 3, 5, 6 of Table 2 are basic.The reason for this is the following.
If (Γ, G) ∈ OG( 4) is basic of independent-cycle type having independent G-unoriented cyclic normal quotients Γ N ∼ = C r and Γ M ∼ = C s , with r ≥ 3 and s ≥ 3 having different parities, then by Theorem 2, (Γ, G) is as described in Rows 3 or 4 of Table 2. Since both quotients are G-unoriented, we could interchange N and M , and r and s, if necessary, and assume that r is odd and hence that (Γ, G) appears in Row 3 of Table 2.
Hence without loss of generality, all basic pairs (Γ, G) ∈ OG(4) of independent-cycle type belong to one of the five families given in Table 4 below.Note that the five rows of this table correspond exactly to the five rows of Table 1 (and of Table 2 apart from row 4).The elements in the column 'Generators' of Table 4 form a generating set for G as given in Table 3 and Definition 2. The subgroups listed in the column labelled 'Normal Subgroups' in Table 4 are notable normal subgroups of the group G which we will use in later arguments.Note also that for all groups G listed in Table 4, all subgroups of the groups in the 'Normal Subgroups' column are also normal in G.This is easily checked by noting the following relations between the various generating elements of these groups: where the result of conjugating by σ depends on whether or not G = G 2 (r, s).
The five possible kinds of basic pairs of independent-cycle type.
We may now prove Theorem 1 by determining for each row of Table 4, the values of (r, s) which produce a basic pair (Γ, G) ∈ OG(4).More precisely, we prove Theorem 1 over the next two sections as follows.First, we show that if (Γ, G) ∈ OG(4) appears in Table 4 and is basic, then (r, s) has one of the values appearing in the corresponding row of Table 1.This therefore is a necessary condition and we prove it in Section 3. Then in Section 4, we show that for each of the values of (r, s) appearing in Table 1, the pair (Γ, G) is in fact basic.

Proof of Theorem 1: Necessary Condition
For the pairs (Γ, G) ∈ OG(4) with independent cyclic normal quotients, that is, those appearing in Table 4, we begin by restricting the possible values of (r, s) which can occur when the pair is basic.We obtain precisely the values of (r, s) given in Theorem 1.The following two lemmas identify these values for r (Lemma 3), and s (Lemma 4).Lemma 3. Suppose that (Γ, G) ∈ OG(4) is as described in one of the rows of Table 4 with appropriate r, s ≥ 3. Suppose further that (Γ, G) is basic of cycle type.Then the following hold:  4 then r ∈ {4, 2p} for some odd prime p.
Proof.(a) Suppose that (Γ, G) is as in Rows 1, 3, or 5. Then M := µ ∼ = C r is a normal subgroup of G, and hence also all subgroups of M are normal in G.If r is a prime then r is odd, since r ≥ 3, and is one of the possibilities in part (a).Suppose now that r is not a prime.Then M contains a proper nontrivial subgroup L. Since L is a normal subgroup of G contained in M , and since M has at least three orbits on Γ, it follows that Γ L is a non-trivial proper G-normal quotient, and hence is a cycle, since (Γ, G) is basic.
Since M is semiregular on V Γ, this implies that µ 2 , and hence µ 2 , lies in L. Then since L = M , it follows that L = µ 2 .Hence µ has even order and so r = 2k for some integer k > 1, and µ 2 has order k.Now consider the subgroup K := µ k of M of order 2. By the same reasoning Γ K is a cycle and K must contain µ 2 , implying that k = 2 and r = 4.This proves part (a).
(b) Suppose now that (Γ, G) is as in Row 2 or 4. Then r is even and M 2 := µ 2 is normal in G, as are each of its subgroups.Also, since r ≥ 3, the subgroup M 2 = 1.Suppose that M 2 is not simple and consider a proper nontrivial subgroup L of M 2 .By the same argument as in case (a), Γ L is a cycle and the vertices (−1, 1) and (1, 1) must lie in the same L-orbit.Again, since M 2 is semiregular on vertices, it follows that µ 2 ∈ L. This is a contradiction, so M 2 ∼ = C r/2 has prime order, meaning that r = 2p where p is a prime.Thus part (b) is proved.Lemma 4. Suppose that (Γ, G) ∈ OG(4) is as described in one of the rows of Table 4 with appropriate r, s ≥ 3. Suppose further that (Γ, G) is basic of cycle type.Then the following hold: (a) If (Γ, G) is as in Rows 1 or 5 of Table 4 then s ∈ {4, q} for some odd prime q.
Proof.Note that in all cases ν and ν 2 are semiregular on vertices.The proofs of (a) and (b) here can thus be handled using virtually identical arguments to the proofs of (a) and (b) of Lemma 3 by swapping the roles of µ and ν, and noting that ν only affects the second coordinate of a vertex (instead of the first).
Before giving our main result for this section we need one more result for the special case where G has a normal subgroup of order 2.
Lemma 5. Suppose that (Γ, G) ∈ OG(4) is as described in one of Rows 1−4 of Table 4 with appropriate r, s ≥ 3.If (Γ, G) is basic of cycle type with L a normal subgroup of G of order 2, then one of r, s is equal to 4.Moreover, in this case L must swap (0, 0) with one of (2, 0), (2, 2), or (0, 2).
Finally, we put together these results to prove Theorem 6 which gives the possible values of (r, s) for Theorem 1. Theorem 6. Suppose that (Γ, G) ∈ OG( 4) is as described in one of the rows of Table 4 with appropriate r, s ≥ 3. Suppose that (Γ, G) is basic of cycle type.Then Γ, G, (r, s) are as in the appropriate Row of Table 5.where p and q are odd primes (possibly equal).Hence there are at most four possibilities in each case.

Row of
We will now show that some of these possibilities cannot occur.For most arguments, we will refer to the conditions on r, s in Table 4.

Proof of Theorem 1: Sufficient Condition
In this section we complete the proof of Theorem 1 by showing that for each of the values of (r, s) given in Table 1, the appropriate pair (Γ, G) from the same table will be basic of cycle type.We do this by showing that, for all such (Γ, G) and (r, s), all of the normal quotients of the graph Γ with respect to minimal normal subgroups of the group G are cycles.
Since all groups G appearing in Table 1 are isomorphic to direct products of cyclic or dihedral groups (or quotients of such groups), we begin with the following lemma concerning minimal normal subgroups of such groups.Proof.Let M be a minimal normal subgroup of G.If M ∩ H = 1 then M ∩ H is a normal subgroup of G and hence must be equal to M by assumption.In particular, M is a minimal normal subgroup of G contained in H.If R = 1 is a normal subgroup of H contained in M , then it is also a normal subgroup of G, so R = M .Hence M must be a minimal normal subgroup of H ∼ = D r and so M ∼ = C p where p is a prime divisor of r.If M ∩ K = 1 then the same argument shows that M is a minimal normal subgroup of K, so M ∼ = C p where p is a prime divisor of s.
Suppose on the other hand that 1 = M ∩H = M ∩K.Then M projects nontrivially onto each of H and K and we claim that M ≤ Z(G).To see this, take m ∈ M where m = h•k where h ∈ H and k ∈ K. Now take an arbitrary g ∈ H and note that since M is normal in G, it follows that m −1 •m g ∈ M .Now since k commutes with all elements in H, we get and since g was arbitrary it follows that h commutes with every element of H.An analogous argument shows that k commutes with each element of K. Hence h ∈ Z(H) and k ∈ Z(K) and so m = hk ∈ Z(H) × Z(K) = Z(G).Therefore M ≤ Z(G).
The conditions M ∩ H = M ∩ K = 1 imply that the projection maps from H × K to H, and from H × K to K, restrict to isomorphisms from M to a subgroup of Z(H), and of Z(K), respectively.In particular both Z(H) and Z(K) are nontrivial.Now Z(D r ) = 1 if r is odd, and Z(D r ) ∼ = C 2 if r is even.Thus r is even, and s is even in this case also.
We now go through each of the rows of Table 4 and show that, for each of the values of (r, s) in Table 5 (which were deduced in Theorem 6), the corresponding pair (Γ, G) appearing in Table 4 is basic of cycle type.The next result deals with the first row of Table 4.
Proposition 8. Suppose that Γ := Γ(r, s) and G := G(r, s) are as described in Row 1 of Table 4.If (r, s) is of the form (4, p), (p, 4), or (p, q) where p and q are odd primes, then (Γ, G) is basic of independent-cycle type.
Proof.By Definition 1(a), G = M × N = µ, σ × ν ∼ = D r × C s , and by [1, Lemma 2.7], Γ N is a G-unoriented cycle and Γ M is a G-oriented cycle, and these are independent cyclic normal quotients.Thus we need only check that (Γ(r, s), G(r, s)) is basic of cycle type.Since one of r, s is odd, Lemma and µ 2 ν 2 .It is also easy to check that Γ + (4, 4) ∼ = K 4,4 .Letting A = {(0, 0), (2, 0), (2, 2), (0, 2)} and B = V Γ + \A denote the bipartition of Γ + , it is clear that each of these three minimal normal subgroups swaps the vertex (0, 0) with some vertex in A. In all three cases the normal quotient with respect to this subgroup is a 4-cycle so (Γ + , G + ) is basic of cycle type.Now suppose that (r, s) is (4, 2p) or (2p, 4), where p is an odd prime.Since G + is an index 2 subgroup of G, each minimal normal subgroup of G + satisfies case (i) or (ii) of Lemma 9.The minimal normal subgroups of G + which are minimal normal in G are precisely M + = µ 2 and N + = ν 2 .This can be checked using Lemma 7 and noting the fact that the central minimal normal subgroups of G(2p, 4) and G(4, 2p) are µ p ν 2 and µ 2 ν p respectively, and neither of these is contained in G + for the reason given in the first paragraph of the proof.Each of Γ + M + and Γ + N + is a cyclic normal quotient of Γ + .
We now show that M + and N + are the only minimal normal subgroups of G + , by showing that G + has no minimal normal subgroups of the form described in case (ii) of Lemma 9. To this end, suppose that L is such a minimal normal subgroup of G + , and that L × L g is normal in G (and contained in G + ) for some g ∈ G\G + .Since G + has order 8p this is only possible if |L| = 2.
Consider an element γ ∈ G + of order 2. Then viewing G as the direct product G = µ, σ × ν , we may write γ = (x, y) where x is equal to 1, µ r/2 , or µ i σ for some i, and y is equal to 1 or ν s/2 .Suppose that L = γ .Since L is normal in G + , we have γ µν = γ, and hence x µ = x.Now (µ i σ) µ = µ i−2 σ = µ i σ since |µ| = r > 2, and it follows that x ∈ {1, µ r/2 }.Moreover, since L = 1 and L is not a minimal normal subgroup of G, it follows that γ = µ r/2 ν s/2 ∈ G + (dropping the direct product notation).Since one of r, s equals 4, it follows that one of µ r/2 , ν s/2 , say z, lies in G + , and hence also zγ ∈ G + .Hence both of µ r/2 , ν s/2 lie in G + , and this is a contradiction since one of r, s equals 2p with p an odd prime.We conclude that M + and N + are the only minimal normal subgroups of G + , and hence that (Γ + , G + ) is basic of cycle type.
In order to produce similar results for rows 3 and 4 of Table 4, we need some preliminary theory on the groups H(r, s).The following lemma describes minimal normal subgroups of the group H(r, s) for certain parameters (r, s).Lemma 11.Let p and q be odd primes, and let H(r, s) = µ, ν 2 , τ σν, τ as in Table 4. Proof.By Definition 1(a), we may view H := H(r, s) as Note that the order of H is 2rs.In cases (a) and (b), µ and ν 2 are minimal normal subgroups of H, while for each of the remaining cases, µ 2 , µ r/2 and ν 2 are minimal normal subgroups of H (of course, µ 2 and µ r/2 are equal if r = 4).Finally, if (r, s) = (2p, 4) then µ p ν 2 is also a minimal normal subgroup of H since it is generated by a central involution.
We will now show that in each of these cases, the minimal normal subgroups of H mentioned above are the only minimal normal subgroups of H.In cases Suppose now that L is a minimal normal subgroup of H which is not minimal normal in H 0 .In this case, L contains a proper subgroup K such that K is minimal normal in H 0 .Then K τ is also a minimal normal subgroup of H τ 0 = H 0 and so K, K τ = K × K τ is a normal subgroup of H. Since K ≤ L and K τ ≤ L τ = L, we have K × K τ ≤ L and K × K τ is normalised by H 0 , τ = H.Hence K × K τ = L, so every minimal normal subgroup of H is either minimal normal in H 0 or is the direct product of two isomorphic minimal normal subgroups of H 0 which are interchanged by τ .It thus suffices to check all minimal normal subgroups of H 0 and determine which of these give rise to minimal normal subgroups of H in this way.
If (r, s) = (p, 4), then H 0 ∼ = C p × C 2 2 and we may check that the minimal normal subgroups of H 0 are µ , ν 2 , K 1 := τ σν and K 2 := τ σν −1 .Note that conjugation by τ interchanges the subgroups K 1 and K 2 ; however K 1 × K 2 contains ν 2 and so K 1 × K 2 is not a minimal normal subgroup of H. Thus µ and ν 2 are the only minimal normal subgroups of H.
If (r, s) = (p, 2q), then H 0 ∼ = C p × D q and, by Lemma 7, the minimal normal subgroups of H 0 are µ and ν 2 and both are minimal normal in H.If (r, s) = (2p, 2q), then H 0 ∼ = C 2p × D q and, by Lemma 7, the minimal normal subgroups of H 0 are µ 2 , µ p and ν 2 and each of these is minimal normal in H.
Finally, in case (e), we have (r, s) = (4, 2p) with |H| = 2rs = 16p.Now letting H 1 := C H (ν 2 ), we see that H 1 ≥ µ, ν 2 , σν −1 .In particular since ν 2 is not central in H, we have 16p In particular, all minimal normal subgroups of H are contained in H 1 .Now notice that since p is odd, we have ν p+1 ∈ H 1 , and so (σν −1 )ν p+1 = σν p ∈ H 1 and this is an involution.Moreover, µ, σν p ∼ = D 4 is normal in H 1 .In particular H 1 = ν 2 × µ, σν p ∼ = C p × D 4 , and so by Lemma 7 the minimal normal subgroups of H 1 are ν 2 and µ 2 .If L is a minimal normal subgroup of H which is not minimal normal in H 1 then again L = K × K τ , for K, K τ isomorphic minimal normal subgroups of H 1 .In particular, no such subgroup L exists since both ν 2 and µ 2 are fixed by conjugation by τ , and so these are the only minimal normal subgroups of H.
In Propositions 12 and 13 we apply the information in Lemma 11 to analyse the pairs (Γ(r, s), H(r, s)) described in Rows 3 and 4 of Table 4.Our proofs rely on [1, Lemma 2.9] which specifies (with M, N # , M + , N + as in Definition 1): • if r is odd then Γ M and Γ N # are independent H(r, s)-unoriented cyclic normal quotients; while • if r is even then Γ M + and Γ N + are independent H + (r, s)-unoriented cyclic normal quotients.
for i ∈ {0, 1, 2, 3}.Of these, only µ 2 , σν p µ and σν p µ 3 are contained in H + .Since conjugation by τ ∈ H + swaps σν p µ and σν p µ 3 , neither of these two involutions generates a normal subgroup of H + , and µ 2 generates the normal subgroup M + of type (i).Thus there are no minimal normal subgroups of H + of type (ii), and we conclude that that (Γ + , H + ) is basic of cycle type.
Finally we deal with Row 5 of Table 4. Proposition 14. Suppose that Γ 2 := Γ 2 (r, s) and G 2 := G 2 (r, s) are as described in Row 5 of Table 4.If (r, s) = (p, q) where p and q are odd primes, then (Γ 2 , G 2 ) is basic of independent-cycle type.
To summarise, we have now completed the proof of Theorem 1: if a pair (Γ, G) ∈ OG( 4) is basic of independent-cycle type, then by Theorem 6, Γ, G, (r, s) satisfy ones of the rows of Table 1.Conversely, it follows from the five propositions in this section that each of these entries does indeed yield a pair which is basic of independent-cycle type.
(a) If (Γ, G) is as in Row 1, 3, or 5 of Table 4 then r ∈ {4, p} for some odd prime p.(b) If (Γ, G) is as in Row 2 or 4 of Table

Lemma 7 .
Suppose G = H × K where H ∼ = D r , and K ∼ = C s or K ∼ = D s for some r, s ≥ 3.If M is a minimal normal subgroup of G then either (a) M is a minimal normal subgroup of H, so ∼ = C p where p is a prime divisor of r; or (b) M is a minimal normal subgroup of K, so M ∼ = C p where p is a prime divisor of s; or (c) both r and s are even, M ≤ Z(G), and M ∼ = C 2 .

Table 1 :
The basic pairs of independent-cycle type.
are as in one of the lines of Table2, and Γ M (and Γ M ) are G-oriented if and only if the entry in column 3 is 'Yes'.