Bitranslations and Symmetric Nets

It is known that every class-regular symmetric μ,m -net is tactical. Also it is known that all μ,m nets with m 2 or μ 1 are tactical. In the work of Al-Kenani and Mavron 2003 , it is proved that every symmetric net with m 3 is tactical if and only if it is class regular. In this paper, we construct 2, 4 -net and show that it is class regular and therefore tactical. New necessary and sufficient conditions are given for a symmetric net to admit a nonidentity bitranslation.


Introduction
A t − v, k, λ design Π is an incidence structure with v points, k points on a block, and any subset of t points is contained in exactly λ blocks, where v > k, λ > 0. The number of blocks is b and the number of blocks on a point is r.
The design Π is resolvable if its blocks can be partitioned into r parallel classes, such that each parallel class partitions the point set of Π. Blocks in the same parallel class are parallel. Clearly each parallel class has m v/k blocks. Π is affine resolvable, or simply affine, if it can be resolved so that any two nonparallel blocks meet in μ points, where μ k/m k 2 /v is constant. Affine 1-designs are also called nets. The dual design of a design Π is denoted by Π . If Π and Π are both affine, we call Π a symmetric net. We use the terminology of Jungnickel 1 see also 2 . In this case, if r > 1, then v b μm 2 and k r μm. That is, Π is an affine 1 − μm 2 , μm, μm design whose dual Π is also affine with the same parameters. For short we call such a symmetric net a μ, m -net.
If Π is a symmetric net, we shall refer to the parallel classes of Π as block classes of Π and to the parallel classes of Π as point classes of Π.
A bitranslation of Π is an automorphism fixing every point and block class which is fixed-point-free or is the identity. It is well known that the bitranslations form a group of order of a factor of m. The bitranslation group has order m if and only if it acts regularly on each point and block class. In which case we say that Π is class regular.

International Journal of Combinatorics
In this paper we find necessary and sufficient conditions for a permutation in S m to induce a bitranslation, by considering regular subsets of S m .
Let Π be a symmetric μ, m -net and let A, B be block classes. If θ : A → B is a bijection, then the point subset S a ∩ θ a , where a ranges over all elements of A, is called an A, B -transversal of Π. If S is a union of point classes of Π, then S is said to be a regular transversal of Π, and θ is called an A, B -syntax of Π. We will denote the set of all A, B -syntaxes by Σ A, B .
Π is defined to be tactical if and only if |Σ A, B | m for all pairs of distinct block classes A and B of Π. Equivalently, the intersection of any two nonparallel blocks is contained in a unique transversal. See 3 for more details.
Label the m blocks in each block class of Π by {1, 2, . . . , m} and similarly for point classes of Π. Then a bijection between point or block classes of Π may be regarded as an element of the symmetric group S m . If A, B are any two distinct block classes of Π, then we may regard Σ A, B as a semiregular subset of S m ; that is, if θ 1 , θ 2 ∈ Σ A, B and θ 1 a θ 2 a for some a ∈ A, then θ 1 θ 2 see 3 .
Let Y be a given block class of Π which we will call the base block class, and let its blocks be labelled We call this labelling the standard labelling, relative to the given base block class Y and point class x.
If Π is a tactical μ, m -net with standard labelling for its block and point classes, then the identity bijection 1 ∈ Σ A, B , for any two block classes A and B of Π.

Bitranslation Groups
Let Π be a symmetric μ, m -net.
Let Y {Y 1 , Y 2 , . . . , Y m } be the base block class of Π and x {x 1 , x 2 , . . . , x m } the base point class of Π in the standard labelling as above. If Φ is a bitranslation of Π, then Φ induces the permutation σ ∈ S m defined by: Then by definition of standard labelling it follows that Φ : K i → K σ i for any point or block class K of Π.
Notation. If X is a subset of a group G, then C G X denotes the centralizer subgroup {g ∈ G | gx xg, for all x ∈ G} of X in G. Assume first that Φ is a bitranslation. Since Φ is a bitranslation, it fixes every point and block classes. Hence Φ fixes S since S is a union of point classes. Therefore Conversely, suppose σ ∈ H. Define Φ as in the statement of the theorem. Clearly Φ is bijective and fixes every point and block classes. Let p j be any point and A i any block of Π.
By definition of standard labelling, p j ∈ Y j , where Y is the base block class. Therefore, Note that θ depends only on p j and A i . Since p j and p σ j are parallel, they are in the same transversal determined by θ: that is, With the notation and hypothesis of the theorem, we prove the following corollaries.
Proof. a The mapping σ → Φ of the theorem is easily verified to be an isomorphism from H onto the bitranslation group. b This follows easily from the definition of class regular.

Regular Subsets
Let n ≥ 2 be an integer and Ω a set of size n. Let S Ω be the symmetric group on Ω.
A subset T of S Ω is a semiregular subset of S Ω if for any α, β ∈ Ω, there exists at most one element t ∈ T such that αt β.
If there exists always exactly one such t ∈ T , then T is a regular subset of S Ω . Suppose T is a regular subset of S Ω .
T , the subgroup generated by T in S Ω . Then it is easy to see that a G is transitive on Ω; b C C S Ω G .
Using this notation, we prove the following results.

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International Journal of Combinatorics Lemma 3.1. If T is regular, then C is semiregular on Ω and |C| divides n.
Let β be any element of Ω and let t ∈ T be such that αt β. Then βx αtx αxt αt β. Therefore, x 1. The rest of the proof is straightforward.
It is clear that T G T if and only if T is a subgroup of S Ω .
b T G. Choose any α ∈ G. For each t ∈ T , define g t ∈ S Ω by If t 1 , t 2 ∈ T and g t 1 g t 2 , then αt −1 1 αt −1 2 and hence t 1 t 2 since T is regular . Therefore, |H| |T | n.
If t 1 , t 2 , u ∈ T , then αu g t 1 g t 2 αt −1 1 u g t 2 αt −1 2 t −1 1 u αg t 1 t 2 . Hence g t 1 g t 2 g t 1 t 2 , which means that the mapping t → g t defines an isomorphism T → H H is just the left regular representation of T . Therefore, |H| |T | n.
Since |H| n and, by Lemma 3.1, |C| divides n, then |C| n and C H.
Lemma 3.3. If g, h ∈ S Ω , then gTh is also a regular set.

Syntax Sets as Regular Subsets
The general theory of regular sets developed in the previous section is now applied syntax sets.
From the introduction to this paper, we know that syntax sets of a tactical symmetric μ, m -net are semiregular subsets of S m . Proof. From Lemma 3.3 we may assume that 1 ∈ T . The order of any of the 3 nonidentity elements of T is therefore either 2 or 4.
Similarly, 2t 4 would imply 4t / 1, 2, 3, or 4, which again is impossible. Therefore, the case 3t 2 is impossible and so 3t 1 as above. If no element of T has order 4, then all nonidentity elements of T have order 2. Then from the regularity of T it follows that T must be the Klein 4-group.
We continue with the notation and hypothesis of Theorem 2.1. 1 Suppose all syntax sets of Π are the same subgroup G of S Ω . Then by Lemma 3.2, H G and the bitranslation group of Π is isomorphic to 2 Consider the special case m 4. By Theorem 4.2, we know that any syntax set of Π is congruent to a subgroup of S 4 of order 4. This must be either the Klein 4-group or one of the 3 cyclic subgroups of order 4. Since Π is tactical, all its syntax sets contain the identity. Therefore, we can say that any syntax set of Π is conjugate to a subgroup of order 4 in S 4 .
Suppose all syntax sets of Π are the same subgroup G. Then by Corollary 2.3, we have H G and the bitranslation group of Π is isomorphic to C S 4 G . From 1 , G ∼ C S 4 G and C S 4 ∩ G G, since G is abelian. Hence C S 4 G G. It follows that the bitranslation group of Π has order 4 and hence Π is class regular.

International Journal of Combinatorics
Below is an example of a tactical symmetric 2, 4 -net in which every syntax set Σ A, B , A / B, is the Klein 4-group: {1, 12 34 , 13 24 , 14 23 }. The full automorphism group has order 5376. The author is grateful to V. D. Tonchev for this information.
The incidence matrix M of this symmetric 2, 4 -net is as follows: