The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Steiner and Kirkman triple systems are undoubtedly amongst the most popular discrete structures. A Steiner triple system of order v, briefly STS(v), is a pair \((V,\mathcal{B})\) where V is a set of v points and \(\mathcal B\) is a set of 3-subsets (blocks) of V with the property that any two distinct points are contained in exactly one block. A Kirkman triple system of order v, briefly KTS(v), is an STS(v) together with a resolution \(\mathcal R\) of its block-set \(\mathcal B\), that is a partition of \(\mathcal{B}\) into classes (parallel classes) each of which is, in its turn, a partition of the point-set V. It has been known since the mid-nineteenth century that a STS(v) exists if and only if \(v\equiv 1\) or 3 (mod 6) [40]. The analogous result for KTSs has been instead solved more than a century later: a KTS(v) exists if and only if \(v\equiv 3\) (mod 6). The first published solution was by Ray-Chauduri and Wilson [53] but the problem was solved at least 8 years earlier by Lu [43] (see [17], p. 13).

An automorphism of a STS is a permutation of its points leaving the block-set invariant. Analogously, an automorphism of a KTS is a permutation of its points leaving the resolution invariant. Thus an automorphism of a KTS is automatically an automorphism of the underlying STS though the converse is not true in general.

For general background on these topics we refer to [18].

In the next section we will give a brief survey of the automorphism groups of STSs and KTSs. Looking at the results it will be evident how little we know about KTSs in comparison with STSs. For instance, it is has been known for close to a century that there is an STS(v) with an automorphism group of order v for any admissible v. Yet, we are still quite far from a similar result for KTSs. At the moment the existence of a KTS(v) with an automorphism group of order v or \(v-1\) is known only when v has a special prime factorization. In particular, there is no known congruence \(v\equiv k\) (mod n) which guarantees the existence of a KTS(v) with an automorphism group of order close to v.

Adopting the combinatorial-analog of the famous Erlangen program by Klein [34], we believe that the interest of a discrete structure is proportional to the number of its automorphisms. Motivated by this and by the shortage of results mentioned above, in this paper we deeply investigate Kirkman triple systems which are 3-pyramidal, i.e., admitting an automorphism group acting sharply transitively on all but three points.

Now we make a short digression to explain why adopting this “philosophy" also brings practical benefits. Given the definition of a discrete structure, the first natural target, which could be very difficult, is to determine under which constraints it exists. The second is to give an explicit construction for this structure; in some cases this could be even more difficult. For instance, thanks to the recent seminal work of Keevash [39], it is known that a Steiner t-design exists provided that the trivial necessary conditions are satisfied and that its order is sufficiently large (the case \(t=2\), due to Wilson [58‐60], dates back to the 70’s). This is an outstanding achievement which seemed completely out of reach only a decade ago; yet, the probabilistic methods used by Keevash are non-constructive, and do not provide an explicit lower bound on the order of a t-design that guarantees its existence. On the contrary, the request to have many symmetries, besides being in compliance with the Erlangen program, allows to develop constructive algebraic methods. This paper gives the complete recipe to construct, explicitly, a KTS(v) for any v as in (i), (ii), (iii) of our main result below.

In particular, (ii) and (iii) allow us to reach our main target of getting some families of highly symmetric KTSs whose orders fill a congruence class.

We point out that in many cases the constructed 3-pyramidal KTSs inherit some automorphisms of the group G acting sharply transitively on all but three points. This permits to increase their number of symmetries considerably (see Remarks 11.2, 11.4 and 11.6).

After the brief survey of Sect. 2 concerning some results on the automorphism groups of Steiner and Kirkman triple systems, the article will be structured as follows. In Sect. 3 we provide the difference methods to construct a 3-pyramidal KTS and we prove that each group having a 3-pyramidal action on a KTS fixes one parallel class and acts transitively on the remaining ones. We also prove that such a group must have exactly three involutions, and these involutions are pairwise conjugate. Groups with this property will be called pertinent. Although in literature there is no lack of articles on groups with three involutions (see, for example, [36, 41]), none of them allows us to determine the set of “relevant” orders. We prove (Theorem 3.9) that such orders are precisely those of the form \(12n + 6\) or \(4^\alpha (24n + 12)\), and from this we partially derive the necessary condition in Theorem 1.1. The proof of the “only if" part of Theorem 3.9 is purely group theoretical and for this reason the whole Sect. 4 is dedicated to it. However, its reading is not necessary for understanding the rest of the article, whose nature is purely combinatorial.

The constructive part of Theorem 1.1 will be proven in Sect. 11 and it is the result of numerous direct constructions (Sects. 6, 7 and the Appendix) and recursive constructions (Sects. 8, 9, 10). These results are preceded by a brief section (Sect. 5) useful for understanding the notation and terminology used throughout the rest of the paper. The recursive constructions required the introduction of new concepts such as doubly disjoint difference family and splittable difference matrix which we believe may be important by themselves.

The article concludes (Sect. 12) with a short list of open problems.

The literature on Steiner and Kirkman triple systems having an automorphism with a prescribed property or an automorphism group with a prescribed action is quite extensive.

For instance the set of values of v for which there exists an STS(v) with an involutory automorphism fixing exactly one point (reversed STS) has been established in [25, 54, 57]: it exists if and only if \(v\equiv 1, 3, 9\) or 19 (mod 24). Results concerning the full automorphism group of a STS have been obtained by Mendelsohn [44] and Lovegrove [42].

Here, we just provide a brief survey of what is known on the existence of systems whose number of automorphisms are at least close to the number of points.

Adopting a terminology coined by Mendelsohn and Rosa [45], we say that a combinatorial design is f-pyramidal if it admits an automorphism group G fixing f points and acting sharply transitively on the others. If \(f=0\) one usually speaks of a regular design and, more specifically, of a cyclic design if the group G is cyclic. If \(f=1\) one usually speaks of a 1-rotational design.

It was proved a long time ago [51] that there exists a cyclic STS(v) for all admissible values of v except \(v=9\). On the other hand the unique STS(9), that is the point-line design associated with the affine plane of order 3, is clearly regular under the action of \(\mathbb {Z}_3^2\). Thus there exists a regular STS(v) for all admissible values of v.

The analogous problem of determining the set of values of v for which there exists a regular KTS(v) is almost completely open and it appears to be very difficult. The few known results on this problem are the following. The parallel classes of the point-line design associated with the n-dimensional affine space over the field of order 3 clearly give a KTS\((3^n)\) that is regular under the action of \(\mathbb {Z}_3^n\). A necessary condition given in [46] for the existence of a cyclic KTS\((6n+3)\) is that \(2n+1\) is not a prime power congruent to 5 (mod 6). This condition is also sufficient up to \(n=32\) [46] and when all prime factors of n are congruent to 1 (mod 6) [31].

The existence of a 1-rotational STS(v) has been thoroughly investigated in [3, 6, 47, 52] leaving the problem open only for the orders v satisfying, simultaneously, the following conditions: \(v=(p^3-p)n+1\equiv 1\) (mod 96) with p a prime; \(n\not \equiv 0\) (mod 4); the odd part of \(v-1\) is square-free and without prime factors \(\equiv 1\) (mod 6).

The 1-rotational KTSs have a very nice structure. Indeed a group with a 1-rotational action on them (necessarily binary, i.e., admitting exactly one involution) is transitive on the parallel classes. On the other hand, as in the regular case, very little is known about their existence which has been proved only for orders v of the the following types: v is a power of 3 (the already mentioned regular KTS\((3^n)\) is also 1-rotational); all prime factors of \({v-1\over 2}\) are congruent to 1 (mod 12) [13]; \(v=8n+1\) with all the prime factors of n congruent to 1 (mod 6) [14].

An f-pyramidal STS(v) with \(f\ne 0\) may exist only for \(f\equiv 1\) or 3 (mod 6) with \(f<v/2\) or \(f=v\) (see Lemma 1.1 in [12]). The existence problem for 3-pyramidal STSs was completely settled in [12].

As an obvious consequence of Theorem 2.1, a 3-pyramidal KTS(v) may exist only when \(v\equiv 9\) (mod 24) or \(v\equiv 15\) (mod 24) or \(v\equiv 3\) (mod 48). The main result of this paper (Theorem 1.1) provides, in particular, a complete answer in the last case \(v\equiv 3\) (mod 48).

Finally, a STS(v) is called 1-transrotational if it has an automorphism group G that fixes exactly one point, switches two points, and acts sharply transitively on the remaining \(v-3\). This terminology was first used in [29] under the assumption that G is cyclic, though G just need to be binary. One cannot fail to notice a certain kinship between 3-pyramidal and 1-transrotational STSs, but apart from the fact that their groups are deeply different, the sets of orders for which they exist do not coincide. Indeed it was proved in [29] that a 1-transrotational STS(v) under the cyclic group exists if and only if \(v\equiv 1, 7, 9\) or 15 (mod 24). It is easy to check that the same holds if we remove the assumption that the group be cyclic. As far as we are aware, nobody studied 1-transrotational KTS(v). Considering the above, they might exist only for \(v\equiv 9\) or 15 (mod 24) but it is not difficult to exclude the case \(v\equiv 15\) (mod 24). This will be shown in a paper in preparation [11] where we will deal with the case \(v\equiv 9\) (mod 24).

In this section we show that the existence of a 3-pyramidal KTS over a group G is equivalent to constructing a suitable difference family (DF) in G relative to a partial spread, a concept introduced by the second author in [6]. We point out that throughout the paper, except for Sect. 4, every group will be denoted additively.

A partial spread of a group G is a set \(\Sigma \) of subgroups of G whose mutual intersections are trivial. If \(\tau =\{d_1^{e_1}, \dots , d_n^{e_n}\}\) is the multiset (written in “exponential" notation) of the orders of all subgroups belonging to \(\Sigma \), we say that \(\Sigma \) is of type \(\tau \) or a \(\tau \)-partial spread. A spread (or partition) of G is a partial spread whose members between them cover the whole group G.

The list of differences of a triple \(B=\{x,y,z\}\) of elements of G is the multiset \(\Delta B\) of size 6 defined byThe list of differences of a family \(\mathcal {F}\) of 3-subsets of G, denoted by \(\Delta \mathcal {F}\), is the multiset union of the lists of differences of all its triples. Also, the flatten of \(\mathcal{F}\), denoted by \(\Phi (\mathcal {F})\), is the multiset union of all the triples of \(\mathcal {F}\).A \((G, \Sigma , 3, 1)\) difference family (DF) is a family \(\mathcal {F}\) of 3-subsets of G (base blocks) whose list of differences is the set of all elements of G not belonging to any member of the partial spread \(\Sigma \). If \(\Sigma =\{H\}\) we write (G, H, 3, 1)-DF or simply (G, 3, 1)-DF when \(H=\{0\}\). If \(\Sigma \) is a partial spread of type \(\tau \), we also use the notation \((G, \tau , 3, 1)\)-DF.

If \(\mathcal{F}\) is a (G, H, 3, 1)-DF, then its size is clearly equal to \({|G\setminus H|\over 6}\) and then its flatten \(\Phi (\mathcal{F})\) has size \({|G\setminus H|\over 2}\). Thus, if J is a subgroup of H of order 2, one can ask whether \(\Phi (\mathcal{F})\) is a complete system of representatives for the left cosets of J that are not contained in H. In the affirmative case we say that \(\mathcal{F}\) is J-resolvable.

We note that the development of a J-resolvable (G, H, 3, 1)-DF is a Kirkman frame [56] admitting G as a sharply point transitive automorphism group.

A multiplier of a J-resolvable (G, H, 3, 1)-DF, say \(\mathcal F\), is an automorphism \(\mu \) of G leaving \(\mathcal F\) invariant. We say that \(\mu \) is a strong multiplier if it fixes H element-wise.

The following fact is straightforward.

Difference families are a crucial topic in Design Theory [2, 17]. In particular, as a special case of Theorem 2.1 in [12], it is possible to characterize the 3-pyramidal STS(\(6n+3\)) in terms of difference families as follows.

In the following lemma we recall what the 3-pyramidal STS\((6n+3)\) generated by a \((G,\{2^3,3^e\},3,1)\)-DF looks like.

Obviously, a pertinent group is necessarily non-abelian. Up to isomorphism, the smallest pertinent group is \(\mathbb {D}_6\), i.e., the dihedral group of order 6. The next one is \(\mathbb {A}_4\), the alternating group of degree 4.

As already said, we prefer to write every group in additive notation. So, differently from the mostly used representation, we prefer to see \(\mathbb {D}_6\) as the additive group D with underlying set \(\mathbb {Z}_2\times \mathbb {Z}_3\) and operation law \( \hat{+}\) defined by \((a,b) \ \hat{+} \ (c,d)=(a+c,(-1)^{c}b+d)\). Adopting this representation, it is easy to see that the difference \( \hat{-}\) in D works as follows:The three involutions of D are (1, 0), (1, 1) and (1, 2). The fact thatconfirms that D is pertinent.

The alternating group \(\mathbb {A}_4\) will be also represented additively as the first term of an infinite series of pertinent additive groups that we will construct in the proof of the “if" part of Theorem 3.9.

The crucial ingredient to characterize and construct the 3-pyramidal KTSs are some special \((G,\{2^3,3\},3,1)\)-DFs with G pertinent defined as follows.

The following fact is straightforward.

Speaking of a \((G,\{2^3,3\},3,1)\)-RDF with G pertinent, we will mean a J-resolvable \((G,\{2^3,3\},3,1)\)-DF with J one of the three subgroups of G of order 2.

The following result gives a characterization of the 3-pyramidal KTSs and, more importantly, a way to construct them.

In view of Theorem 3.8, it is important to determine the set of pertinent numbers, i.e., the set of orders of the pertinent groups.

The alternating group \(\mathbb {A}_4\) can be seen, up to isomorphism, as the group \(G_1\).

Here we prove the “only if" part of Theorem 3.9. Considering that the arguments used are purely group theoretical, the reading of this section can be postponed to a later time without compromising the understanding of the rest of the article. Also, we point out that in this section, unlike the rest of the paper, we prefer to denote groups in multiplicative notation.

In the following, let G be a pertinent group and denote by K the subgroup of G generated by the three involutions i, j, k. Let \(C_G(K)\) be the centralizer of K in G, that isand set \(C=C_G(K)\). Clearly, K is a characteristic subgroup of G, so both K and C are normal in G. Since G acts on \(\{i,j,k\}\) by conjugation with kernel C, the quotient G/C is isomorphic to a transitive subgroup of \(S_3\), so either \(G/C \cong S_3\) or \(G/C \cong A_3\) (here \(A_3\) denotes the alternating group of degree 3). Observe in particular that G has an element of order a power of 3 acting “cyclically” on the involutions (meaning that it sends i to j, j to k and k to i). Recall that if H is a subgroup of G the “normalizer” of H in G is \(N_G(H)=\{g \in G\ :\ H^g=H\}\), where \(H^g=g^{-1}Hg\) is a “conjugate” of H in G.

We will make use of the following well-known result in group theory.

We start providing sufficient conditions for a pertinent group to have subgroups or quotients that are pertinent.

For a pertinent group of order \(2^n \cdot d\) with d odd, the following two lemmas give us sufficient or necessary conditions for n to be even or odd.

To prove the main result of this section, we need the following result.

We are now ready to prove the “only if" part of Theorem 3.9.

A maximal prime power divisor of any integer n will be called a component of n. As it is standard, given a prime power q, we denote by \(\mathbb {F}_q\) the field of order q. We extend this notation to any integer \(n>1\) denoting by \(\mathbb {F}_n\) the ring which is direct product of all the fields whose orders are the components of n. Thus, for instance, \(\mathbb {F}_{45}=\mathbb {F}_5\times \mathbb {F}_9\). The additive group of the ring \(\mathbb {F}_n\) will be denoted by \(V_n\) and we set \(V_n^*:=V_n\setminus \{0\}\). If \(n=1\), then \(V_n\) is the trivial group with one element. If d is a divisor of n, any subgroup S of \(V_n\) of order d is clearly isomorphic to \(V_d\) and therefore, by abuse of notation, such a subgroup S will be often denoted by \(V_d\).

The group of units of \(\mathbb {F}_n\) will be denoted by \(\mathbb {U}(\mathbb {F}_n)\) and its order by \(\psi (n)\). Obviously, in the particular case that \(n=q\) is a prime power, \(\mathbb {U}(\mathbb {F}_n)\) is nothing but the multiplicative group \(\mathbb {F}_q^*\) of the field \(\mathbb {F}_q\) and \(\psi (n)=q-1\). Otherwise, if n has more than one component, say \(q_1, \dots , q_\omega \), then \(\mathbb {U}(\mathbb {F}_n)=\mathbb {F}_{q_1}^* \times \dots \times \mathbb {F}_{q_\omega }^*\) and \(\psi (n)=\prod _{i=1}^\omega (q_i-1)\). The set of non-zero squares and of non-squares of the field \(\mathbb {F}_q\) will be denoted by \(\mathbb {F}_q^\Box \) and , respectively.

If A, B are non-empty subsets of \(\mathbb {F}_n\), then AB will denote the multiset \(\{ab \ | \ a\in A; b\in B\}\). If \(A=\{a\}\) or \(B=\{b\}\), then AB will be written as aB or Ab, respectively.

Let \(q_1\), ..., \(q_\omega \) be the components of an odd integer n. For every non-empty I belonging to the power-set \(2^{\{1,\dots ,\omega \}}\), choose an element \(c(I)\in I\) and consider the subset S(I) of \(V_n^*\) defined as follows:Then define \(S:=\displaystyle \bigcup _{I\in 2^{\{1,\dots ,\omega \}}\setminus \{\emptyset \}} S(I)\). Such a set S has size \({n-1\over 2}\) and then will be called a halving of \(V_n^*\). It is easy to see that it has the following property:Given a group G and an element s of \(\mathbb {F}_n\), the endomorphism of \(G\times V_n\) mapping (x, y) to (x, ys) will be denoted by \(\mu _s\). It is evident that if \(s\in \mathbb {U}(\mathbb {F}_n)\), then \(\mu _s\) is an automorphism of \(G\times V_n\).

Given \(\alpha \ge 1\), remember that throughout the paper \(G_\alpha \) will denote the group defined in the “if" part of Theorem 3.9. By canonical involution of \(G_\alpha \) we will mean \((0,2^{\alpha -1},2^{\alpha -1})\). Also, the canonical involution of \(G_\alpha \times V_n\) will be \((0,2^{\alpha -1},2^{\alpha -1},0)\). Speaking of a (G, H, 3, 1)-RDF with \(G=G_\alpha \) or \(G=G_\alpha \times V_n\), we will mean a \(\{0,j\}\)-resolvable \((G_\alpha ,H,3,1)\)-DF where j is the canonical involution of G.

The five smallest pertinent values of n are 6, 12, 30, 36 and 48. In the following, for each of these values, a 3-pyramidal KTS\((n+3)\) will be given by means of a \((G,\{2^3,3\},3,1)\)-RDF with \(G=D\), \(G_1\), \(D\times V_5\), \(G_1\times V_3\) and \(G_2\), respectively. By way of illustration, in the first two cases we follow the instructions of Theorem 3.8 and we concretely construct a 3-pyramidal KTS(9) and a 3-pyramidal KTS(15).

The realizations of these five small KTSs allow us to state the following.

Proposition 6.1 will enable us to construct infinite classes of 3-pyramidal KTSs in the subsequent sections.

The action of a group U on a set V is said to be semiregular if the non-identity elements of U do not fix any element of V. The following fact is straightforward.

We need the following lemma.

Although the notion of a SDF was implicitly used in the literature for a long time, the formal definition has been given in [5]. Since then, SDFs have been crucial for the construction of various combinatorial designs in several papers such as [9, 10, 15, 19, 20, 48, 61]. As far as we are aware, the notion of a J-resolvable SDF is new.

Observe that the set of initial base blocks \(\mathcal{B}\) considered in Theorem 7.4 is a lifting of a J-resolvable (D, 3, 4)-SDF.

If we apply Theorem 7.4 with \(n=1\) we are forced to take \(u=3\) and one can see that the resultant RDF is exactly the one given in Subsect. 6.3.

Analogously to Theorem 7.4 the set of initial base blocks \(\mathcal{B}\) considered above is a lifting of a J-resolvable \((G_1,3,4)\)-SDF.

This time the set of initial base blocks \(\mathcal{B}\) considered above is a lifting of a J-resolvable \((G_1,3,6)\)-SDF.

Now we define a class of \((G,\{2^3,3\},3,1)\)-DFs, that we call pseudo-resolvable, that will be crucial for the construction of a 3-pyramidal KTS(v) with \(v=36n+3\) or \(v=48n+3\) or \(v=108n+3\) with all the components of n congruent to 7 or 11 (mod 12).

Even though the definitions of resolvable and pseudo-resolvable difference families are very similar, they are independent. For instance, in spite of the fact that there exists a \((G_1,\{2^3,3\},3,1)\)-RDF (see Subsect. 6.2), there is no \((G_1,\{2^3,3\},3,1)\)-PRDF. Certainly, a \((G,\{2^3,3\},3,1)\)-DF cannot be resolvable and pseudo-resolvable at the same time. Here is a useful example in the groups \(G_1\times V_3\).

The following Theorem 8.3 explains why pseudo-resolvable DFs can be helpful in the construction of some resolvable DFs.

Letting \(G=G_1\times V_3\) and applying Theorem 8.3 to the \((G,\Sigma ,3,1)\)-PRDF given in Example 8.2, we obtain the following important result.

In Appendices A and B we will give an example of a \((G,\{2^3,3\},3,1)\)-PRDF with both \(G=G_1\times V_9\) and \(G=G_2\). Thus, as another important application of Theorem 8.3 we get the following.

A (G, H, 3, 1)-DF is said to be disjoint if its blocks are pairwise disjoint and do not meet the subgroup H. It is well known that there exists a disjoint \((\mathbb {Z}_{6n+1},3,1)\)-DF and a disjoint \((\mathbb {Z}_{6n+3},\{0,2n+1,4n+2\},3,1)\)-DF for every positive integer n (see [8, 22, 23]). There is an intriguing conjecture by Novák [49] according to which every cyclic STS\((6n+1)\) is generated by a suitable disjoint \((\mathbb {Z}_{6n+1},3,1)\)-DF. In a very recent paper [28] it has been proved that this conjecture is true with the possible exception of finitely many composite orders \(6n+1\).

Let us say that two difference families are strongly equivalent if each block of the first is a suitable translate of a block of the second.²

Note how Definition 9.1 reminds a bit of the notion introduced in [21] of a \((v,k,\lambda )\) tiling of a group G, that is a set of mutually disjoint \((v,k,\lambda )\) difference sets in G partitioning \(G\setminus \{0\}\). Needless to say, however, that two distinct members of a \((v,k,\lambda )\) tiling of G cannot be strongly equivalent.

Note that any \(\{0,j\}\)-resolvable (G, H, 3, 1)-DF, say \(\mathcal{F}=\{B_1,\dots ,B_n\}\), is doubly disjoint. Indeed, setting \(B'_i=B_i+j\) for \(i=1,\dots , n\), it is clear that \(\mathcal{F}\) and \(\mathcal{F}':=\{B'_1,\dots ,B'_n\}\) are strongly equivalent. Also, by definition, we have \(\Phi (\mathcal{F})+\{0,j\}=G\setminus H\) and this is equivalent to say that the blocks of \(\mathcal{F}\) and \(\mathcal{F}'\) form a partition of \(G\setminus H\).

In the next section we will give a composition construction for RDFs where doubly disjoint difference families play a crucial role. For this reason, it is worth studying their possible existence. In particular, we are interested in \((\mathbb {Z}_3\times V_{2n+1},\mathbb {Z}_3\times V_1,3,1)\)-DFs. We are going to prove their existence in the case that all the components of \(2n+1\) are congruent to 1 (mod 4).

We recall that a (h, k, 1) difference matrix in a group H of order h, briefly denoted by (H, k, 1)-DM, is a \(k\times h\) matrix with elements from H in which the difference of any two distinct rows is a permutation of H.

In particular, an (H, 3, 1)-DM is equivalent to a complete mapping of H. There is a large literature on complete mappings starting with the famous conjecture of Hall and Paige [32] according to which a group H of even order admits a complete mapping if and only if it is admissible, i.e., if and only if its 2-Sylow subgroups are not cyclic. The conjecture has been finally proved in [26].

It is quite evident that there exists a homogeneous (H, k, 1)-DM if and only if there exists an \((H,k+1,1)\)-DM. Indeed, adding a null-row to a homogeneous (H, k, 1)-DM one gets an \((H,k+1,1)\)-DM. Conversely, if M is a \((H,k+1,1)\)-DM with rows \(M_1\), ..., \(M_{k+1}\), then one gets a homogeneous (H, k, 1)-DM whose rows are \(M_2-M_1\), ..., \(M_{k+1}-M_1\).

Thus, as immediate consequence of the main results on (H, 4, 1)-DM (see [30, 50]), we have the following.

Difference matrices are also a crucial topic of Design Theory [2, 17]. They have been used explicitly or implicitly in a lot of papers especially for the composition constructions of designs with a regular automorphism group starting from some early work by Jungnickel [37] and by Colbourn and Colbourn [16]. Homogeneous difference matrices have been used later for the composition constructions of several kinds of resolvable designs (see, e.g., [1]). As far as we are aware the quite appropriate term homogeneous was coined in [38] when other authors choose other terms as good [13] about the same time.

We give three examples of splittable difference matrices that will be crucial for the construction of some classes of resolvable difference families.

The matrix of the third example is also homogeneous. The following Theorem 10.8 explains how doubly disjoint or resolvable difference families in a quotient group G/H can be combined with homogeneous or splittable difference matrices in H for the construction of resolvable difference families in G.

As an important consequence of Theorem 10.8 we get the following corollaries.

We are finally able to prove the sufficient conditions given by the main Theorem 1.1.

The problem of classifying the 3-pyramidal KTS(v) remains open in the following cases.The open cases above could be closed if one solves the following problems, respectively.

Our research naturally leads to consider also the following collateral problems which, in our opinion, are interesting on their own.