Steiner Loops of Affine Type

Steiner loops of affine type are associated to arbitrary Steiner triple systems. They behave to elementary abelian 3-groups as arbitrary Steiner Triple Systems behave to affine geometries over GF(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {GF}}(3)$$\end{document}. We investigate algebraic and geometric properties of these loops often in connection to configurations. Steiner loops of affine type, as extensions of normal subloops by factor loops, are studied. We prove that the multiplication group of every Steiner loop of affine type with n elements is contained in the alternating group An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_n$$\end{document} and we give conditions for those loops having An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_n$$\end{document} as their multiplication groups (and hence for the loops being simple).


Introduction
A quasigroup is a set L of elements endowed with a binary operation (•) which does not need to be associative, and which is such that the equations a • x = b and y • a = b determine unique solutions x = a\b and y = b/a, thus its multiplication table is a latin square, and a loop is a quasigroup which has a neutral element Ω (see [28] and [11] for a general reference).
As pointed out first by R. Baer in [5], Th. 1.1., Corollary 1.2, loops can be described as homogeneous spaces of their multiplication group G L , generated by the left (resp. right) translations λ a : x → a • x (resp. ρ a : y → y • a): if π : G L −→ L , g → gΩ is the orbit map, then the stabilizer H of Ω in G L is a subgroup containing no non-trivial normal subgroup of G L . The section σ : L → G L , x → λ x , of π is such that: (i) σ(L ) is a transversal of H in G L , (ii) σ(Ω) = id ∈ G L , (iii) σ(L ) generates G L , and iv) for every x, y ∈ L there exists precisely one λ z ∈ σ(L ) such that λ z (x) = y (cf. also [25], Sect. 1.2).
If L 1 (•) and L 2 ( * ) are loops, a triple of bijections (α, β, γ) : L 1 −→ L 2 such that α(x) * β(y) = γ(x • y) is called an isotopism. As the only basic property for a loop is that its left and right multiplications λ a : x → a • x and ρ a : x → x • a are bijections, loop theory is often developed up to isotopisms, Vol. 75 (2020) Steiner Loops of Affine Type Page 3 of 25 148 instead of isomorphisms (which, in particular, turn out to be isotopisms where α = β = γ). Kernels N of homomorphisms turn out to be defined by the relations for any x, y ∈ L , which define normal subloops, and which, in the case of a commutative loop L (+) reduce to the only x + (y + N ) = (x + y) + N (1) (cf. [29], p. 1189). A normal subloop N determines a partition of the loop L in cosets (notice that this is not necessary for arbitrary subloops), thus |L | = |N | · |L /N |. If a loop L has only its trivial normal subloops {Ω} and L, then it is called simple. In the case where L is a simple loop, the multiplication group of L turns out to be primitive (see [1], Th. 8, p. 516). A Steiner triple system is a pair (S , T ), where T is a family of triples of elements of S such that any two elements of S are contained exactly in one triple of the family T . Steiner triple systems exist if and only if n is equivalent to 1 or 3 modulo 6. We will use the abbreviation STS or STS(n) for a Steiner triple system, or for a Steiner triple system on n elements, respectively. Since the mid-twentieth century, a large literature has been devoted to Steiner triple systems (a fundamental monograph on the subject is [12]). Basic examples of STS's include the point-line geometry of any projective space over GF (2) or the one of any affine space over GF (3), which for n = 7 and n = 9 are the only STS(n) (for our convenience, we take the occasion to recall that the unique STS (7) is usually called the Fano plane). If any two distinct intersecting triples are such that the minimal Steiner triple subsystem containing them is the STS(9), then the STS is called a Hall triple system (HTS). An isomorphism f : S 1 −→ S 2 of Steiner triple systems (S 1 , T 1 ) and (S 2 , T 2 ) is a bijective map which moves the triples of T 1 onto triples of T 2 .
In the present paper, we study a family of commutative loops of exponent 3 associated to STS's, which are defined by the following operation: Definition 1. Let S be a Steiner triple system, and let Ω ∈ S be fixed. For each x ∈ S , define its opposite −x as the third point μ(x) in the triple {x, Ω, μ(x)} through x and Ω, define the addition x + Ω = x, x + x = −x, and, for any y = x ∈ S \ {Ω}, where z is the third point in the triple through x, y.
With the above defined operation, the Steiner triple system S turns into a loop L S , that we will call Steiner loop of affine type with identity Ω, where the triples {x, y, z} of (S , T ) are characterized by the property that We stress the fact that for any triple in T the associative property holds. The Steiner loop of affine type associated to a STS is a group exactly in the case where the STS is an affine geometry AG(d, 3) of the affine lines over the Galois field GF(3).
Remark 1. Note that, if L 1 and L 2 are the commutative Steiner loops of affine type associated to the same STS by fixing two different elements Ω 1 and Ω 2 , and if we denote the opposite maps by μ 1 and μ 2 , according to the triples {x, Ω 1 , μ 1 (x)} and {x, Ω 2 , μ 2 (x)} in T , then the map γ : If S is a HTS, then any two Steiner loops of affine type associated to S by fixing two different identities Ω 1 and Ω 2 , are even isomorphic (see Remark 2), but if S is not a HTS, then this is in general not true: the two Steiner loops of affine type with identity 1 and 2, defined by the following addition tables can be shown, after tedious computations, to be non-isomorphic, still isotopic and defining the same STS(7).

Remark 2.
(i) In Definition 1 we denoted with −b the third element in the triple containing b and Ω, so that −b + b = Ω. Since, usually, a − b denotes in Loop theory the solution of the equation x + b = a, and since this turns to be identically equal to a + (−b) (if and) only if the Steiner loop of affine type has the inverse property (cf. Theorem 5) we have to stress that, to avoid ambiguity, we will always write −b + a or, when this is not possible, we will write a + (−b). We thank the referee for suggesting this more exact notation.
(ii) The addition + given above in Definition 1 is the same as the second addition • given by Chein in [11], II.9.9 Example, p. 86. In this paper, Chein gives two different additions, which turn to coincide precisely in the case where the STS is a Hall triple system, as we remark here. For Hall triple systems, however, this other addition * was already given in the 1960's by M. Hall and R. H. Bruck, who put, in fact, x * y = z, where z is the third point in the triple through the two elements lying, respectively, in the triple through Ω and x and in the triple through Ω and y, that is, in our notation (see Proposition 3), For a Hall triple system S , the Steiner loop of affine type L S turns out to be a commutative Moufang loop of exponent 3 [20] (see also [16]), i.e. the sequence  [7], any two Steiner loops of affine type associated to a HTS by fixing two different identities Ω 1 and Ω 2 , are isomorphic.
Whereas the loops of affine type associated to Hall triple systems fulfill Moufang's identity general Steiner loops of affine type only fulfill which is obtained by the former when one takes z = y (cf. [3]).

Remark 3.
To any STS one can associate another commutative loop. In fact, already in 1958, Bruck remarked that a commutative totally symmetric loop, that is, such that x • (x • y) = y is essentially an algebraic version of a Steiner triple system, where the triples are exactly the sets {x, y, x•y}, and the identity Ω is an extra point, not in the STS [7]. These other loops have been studied, among the others, in [29] and in [30]. Also in such totally symmetric loop, the triples {x, y, z} of (S , T ) are characterized by the property that, however we associate the product, the equality holds (and in fact we came to the present paper after a series [9], [10], [14], [15], concerned with this characterization of blocks in a design). Such Steiner loop associated to a STS is a group exactly in the case where the triples of the STS are the projective lines of a projective geometry PG(d, 2) over the Galois field GF(2).
For any a, b ∈ S the solution x ∈ S of the equation a + x = b is x = −(−b + a). As stressed in Remark 2 (i), we will see in Sect. 3 that −(−b + a) = −a + b if and only if a, −b and Ω form a mitre centered in Ω.
As remarked already in II.9.9 Example in [11], p. 86, our definition yields a commutative loop L S of exponent 3 fulfilling the weak inverse property (introduced in [27]), that is, such that for any 3-sets {x, y, z} of L S with x + y + z = Ω the associative property holds. Conversely, it is easy to see that, if L S is a commutative loop fulfilling the weak inverse property such that 3x = Ω, then L S gives the structure of a Steiner triple system S L . Obviously, S = S L S and L = L S L .
Since the left translation map from L S to L S , λ a : x → a+x, coincides, for all a ∈ L S , with the right translation map ρ a : x → x + a, the group generated by all left translations of L S coincides with the group generated by all right translations of L S and we denote it by Mult(L S ).

General Facts
In this section we settle the basic correspondences between Steiner triple systems and their associated Steiner loops of affine type. Proof. (i) R is a subsystem of S containing Ω if and only if L R is closed under the operation of L S , hence the assertion follows.
hence, for any ω 1 , ω 2 ∈ L R the third element in the triple through x + ω 1 and x+ω 2 belongs to x+L R , that is, x+L R is a Steiner triple subsystem of S . (iii) This follows from the fact that Q is a commutative loop of exponent 3 with the weak inverse property.

Remark 4.
It must be noticed that, in the case where L R is a normal subloop of L S , the corresponding cosets are not necessarily isomorphic Steiner triple subsystems. This will be manifest in the case 4.3 described in Sect. 4, where L R is an affine hyperplane.
where each τ i is an even permutation of the form Proof. Each λ x can be written as where σ x is a permutation without any fixed point on the set As σ x has no fixed point it does not contain any 1-cycle.
, then the STS has the following blocks: it is an even permutation. Then also every λ x = (Ω, x, −x)σ x is an even permutation.
Remark 5. For our convenience, we remark that Mult(L S ) is contained in A n , and the stabilizer Stab

Computing within a Steiner Loop of Affine Type
Steiner triple systems are often studied through their configurations, which are given subsets of triples. Starting from two triples {z, a 1 , a 2 } and {z, b 1 , b 2 } through one point z, two cases are possible: either the third point c 1 in the triple through a 1 and b 1 coincides with the third point c 2 in the triple through a 2 and b 2 (Pasch configuration C 16 ), or not (configuration C 14 ). In the latter case, one can distinguish further the case where the triple containing the two points c 1 and c 2 contains also z (the mitre configuration centered in z) and the case where it does not (configuration C A ). The Pasch configuration (that is, the 4-triples configuration on the lefthand side in Fig. 1) and the mitre centered at the point Ω (that is, the 5-triples configuration on the right-hand side in Fig. 1), which are those appearing in the projective geometry PG(d, 2) and in the affine geometry AG(d, 3), respectively, are the two most commonly studied configurations.
According to the well-known axioms by Veblen and Young [31] (see [12], p. 147), a point x is called a Veblen point, if any two triples through x determine a Pasch configuration (hence a Fano plane). Similarly, in 3.2 we call a point z a Hall point, if any two triples through z determine a mitre centered at z. Thus the map ι x , defined in (2), plays a role in distinguishing the case where Ω is contained in a Pasch configuration (see Proposition 1) from the case where Ω is a Hall point (see Proposition 5).

Veblen Points in Steiner Loops of Affine Type
We begin by pointing out the role played by the map ι x when Ω is contained in a Pasch configuration: Proof. The first assertion is just an algebraic description of Fig. 1. As for the second one, it is sufficient to note that, with the notations in Fig. 1, For the other direction, under the assumption that λ 2 x (y) = y and ι x (y) = −y, we have to show that {x + y, x, −y} and {x + y, −x, y} are blocks. We have Ω = −y + y = ι x (y) + y = − x + (y + x) + y and Ω = −y + y = −y + λ 2 x (y) = −y + x + (x + y) , and the assertion follows.
As mentioned above, for Steiner triple systems, the concept of Veblen point reduces to the following (see [12], p. 147): Definition 2. Let x be a point in a Steiner triple system S . If for any y, p 1 , p 2 , p 3 , p 4 ∈ S such that y = x and together with {x, p 1 , p 3 }, {x, p 2 , p 4 }, {y, p 1 , p 2 }, also {y, p 3 , p 4 } is a triple of S , then x is called a Veblen point. Alternatively, the point x is called a Veblen point, if any two triples through x determine a Fano plane (Fig. 2).
The following theorem will be used in Theorem 12 to prove that the multiplication group of a simple Steiner loop of affine type, corresponding to a STS(n) containing a Veblen point, is the alternating group on n letters.
Proof. (i) If Ω is a Veblen point, then any two triples through Ω generate a Fano plane and this yields the assertions in (i). Conversely, if λ x and λ −x have the form as in (i), then is equivalent to the assumption that ι x fixes the points x, −x, and Ω, and interchanges y and −y, for any y = x.

Hall Points in Steiner Loops of Affine Type
In this section we consider the mitre configuration within the frame of Steiner loops of affine type. The following straightforward proposition has already been mentioned in the above Remark 2, where we wrote that for HTS's the two definitions given by Chein coincide with the one given by Hall and Bruck.   (Fig. 3).

Recall that a loop (L , * ) fulfills the inverse property if there exists a bijective map
The following theorem demonstrates a connection between IP loops and mitre configurations. Proof. If ι x = id for any x ∈ S , then y = ι x (y) = −x + (x + y), which is precisely the right inverse property of L S .
If L S has the inverse property, that is, y = −x + (x + y) for all x = y in S , but this means that the third point in the triple through −x and x + y is −y, which proves the assertion.
Finally, if Ω is a Hall point, then ι x = id for any x ∈ S by Proposition 3.

Corollary 1. If S is a Hall triple system, then x → −x is an automorphism of L S , hence L S has the inverse property.
Proof. The assertion follows from the fact that, in a HTS, every point is a Hall point (cf. [22]), and we can apply Proposition 3 and Theorem 5.
Proof. If R ≤ S is a Steiner triple subsystem of order 9 containing the triple T 0 = {−x, Ω, x}, then L R is the elementary abelian group of cardinality 9 and the triples are exactly the 3-subsets of elements summing up to zero. In particular, the two parallel triples to T 0 = {−x, Ω, x} are λ y (T 0 ) and λ −y (T 0 ), for a chosen y ∈ R \ T 0 . Thus the restriction of λ x to R is simply ). The claim follows inductively.
Remark 7. Note that, by Theorem 3, the condition on λ x in Theorem 6 is equivalent to saying that λ x has order three.
Remark 8. By a well known theorem, proved first by B. Fisher, thereafter by many others (see [18], cf. also Theorem 8 in the present paper), the order of any Hall triple system is n = 3 k , hence the number 2t + 1 of 3-cycles in Theorem 6 is obtained for t = 3 k−1 −1 2 .

Theorem 7. Let S be a STS and let L S be the corresponding Steiner loop of affine type with identity element Ω. Each left translation λ x is an automorphism of S if and only if S is a Hall triple system.
Proof. Let S be a HTS, let x = Ω be in S , and let {a, b, c} be a triple. If Conversely, assume that each left translation λ x is an automorphism of the STS S . Let {a, b, c} be a triple and x an element not contained in it. We have to prove that {a, b, c} and x generate a STS (9).
Finally, let T 1 = {−(a + x), c, m} and T 2 = {−(c + x), a, n} be triples, and apply λ x to T 2 to produce the triple + x), c, m}, we find that n + x = −m, because of the argument that the opposite of a triple is also a triple. This proves that {a, b, c} and x are contained in a STS (9).
Note that, since S is a HTS, this can be written also as f (z) = f 0 (z) + f (Ω).
The assertion that f 0 is an automorphism of L S follows because the map f 0 = λ −f (Ω) • f is (again) an automorphism of S , such that f 0 (Ω) = Ω and we can apply Theorem 2.
Conversely, for any automorphism f 0 of L S , and for any given ω ∈ S , the map f (z) = f 0 (z)+ω is an automorphism of S simply because f = λ ω •f 0 , and we apply Theorems 2 and 7.

Extensions of Steiner Loops of Affine Type
In this section we reduce the structure of Steiner loops of affine type to consecutive extensions of simple ones. As one can expect, by considering that the number of STS's with n elements increases as n/e 2 + o(n) n 2 /6 (see [21]), this construction is very flexible, compared with the corresponding extension theory for commutative groups. Proof. The first result is standard. The second one follows from the fact that the commutative Moufang loop of exponent 3 corresponding to a HTS always contains a proper normal subloop, namely the associative center Z(L S ) = {z : ∀x, y ∈ L S , (x + y) + z = x + (y + z)} (see [6]), thus we can make a recursive argument on Z(L S ) and L S /Z(L S ).

Theorem 8. Any Steiner loop L S of affine type has a subnormal series
Extensions of normal (sub-)loops N by (quotient) loops Q are much more relaxed than in the case of groups (cf. [2] and [8,24]), simply because for Steiner loops of affine type the associativity holds only for elements x, y and z such that x + y + z = Ω (where possibly x = y = z). We illustrate this in Example 4.1, after giving the following definition, where we brief these conditions for a Steiner loop of affine type to be such an extension (for our convenience, here we will denote the identity elements by 0).

Definition 4.
Let N and Q be Steiner loops of affine type of order n and m, and identity elements 0 and0, respectively, and let Q(N ) be the set of n × n latin squares with coefficients in the set N .
An operator Φ : Q × Q −→ Q(N ), which maps the pair (x,ȳ) to a latin square Φx ,ȳ : N ×N −→ N , is called a Steiner operator if it fulfills the following conditions: (i) the latin square Φ0 ,0 is the (symmetric) table of addition of N ; , for all (x, x ), (ȳ, y ) ∈ Q × N , for all z ∈ N , and forx +ȳ +z =0.
Remark 10. We want to stress the fact that the conditions in the above definition are weak.
-This is manifest for conditions (i) and (ii).
-Forx =0, conditions (iii), (iv) and (v) hold, simply because Φ0 ,0 is the table of addition of a commutative loop of exponent 3. -Forx =0, once arbitrary latin squares Φx ,x and Φ −x,−x are rearranged in such a way to fulfill condition iv), which will determine the same main diagonal of these latin squares, condition v) determines only the places of the element 0 in each row of the otherwise arbitrary latin square Φx ,−x . On the contrary, once the latin squares Φx ,x , Φ −x,−x , and Φx ,−x are chosen, conditions iv) and v) determine now the whole latin squares Φx ,0 and Φ0 ,−x , as we show in Example 4.1, working on the table Vol. 75 (2020) where we denote, with abuse of notation, x ⊕ y = Φx ,ȳ (x , y ), then L is a Steiner loop of affine type of order v = nm, with identity (0, 0 ), such that the opposite of the element ( The subloopN containing all the elements (0, x ), x ∈ N , is a normal subloop, isomorphic to N , and such that L /N is isomorphic to Q. Conversely, any Steiner loop L of affine type, having a normal subloop N and a quotient loop Q = L /N , is isomorphic, for some given Steiner operator Φ, to the above one.
If (ā, a ) and (b, b ) are two given elements in L , then the equation Finally, we want to give an example and to examine some special extensions.

Example
We sketch the construction of the addition table of a Steiner loop L of affine type with 21 elements as an extension of a normal subloop N with 3 elements with a Steiner quotient loop Q of affine type with 7 elements, for which we fix the following addition tables: N : The elements of L will be represented by pairs (x, x ) in Q × N and the addition table of L will be given in 3 × 3 block matrices Φī ,j L : . This proves that S N is an affine hyperplane of S containing Ω, because the cosets of a normal subloop are subsystems according to Theorem 1 (ii).

Corollary 3. If L S is simple, then S does not contain an affine hyperplane.
If L S is not simple, and N is a proper normal subloop of L S , then N is an affine hyperplane of the subloop M generated by N and x, for any x ∈ L \ N .
Proof. The first assertion is trivial, and the second follows from the fact that M turns out to be the union of N , x + N and −x + N .

Quasidirect Sums
Given two STS's R and T , we can build in a standard way an STS S , that we call the quasidirect sum of R by T , in the following way: let R −→ Top(T ), x → ψx, be a map from R to the group of isotopisms of T {(x, x ), (ȳ, y ), (z, z )} is a triple, if and only if, {x,ȳ,z} is a triple of R, and ψx(x ), ψȳ(y ), ψz(z ) is a triple of T .
In the case where ψx = id for allx ∈ R, we obtain the direct sum of R and T .
By Theorem 9, our investigation is confined to simple Steiner loops of affine type.

Simple Steiner Loops of Affine Type
In this final part of the paper we describe the situation in the case when S is a simple STS, that is, the case where L S is a simple loop. We recall that in this case, the group Mult(L S ) is primitive. Moreover, we recall that this group plays a fundamental role, since that the orbit-map Mult(L S ) −→ L S defined by is a loop homomorphism whose kernel is the stabilizer of Ω.
Write σ x = λ x (Ω, x, −x) −1 and denote by Σ the group generated by the set {σ x : x ∈ L S } of permutations, thus each σ x fixes exactly the three elements Ω, x and −x, hence Σ is contained in the stabilizer of Ω. By Theorem 3, the group Mult(L S ) is contained in the alternating group A n , thus Σ is contained in a subgroup isomorphic to the alternating group A n−1 . In [30] it is proved that if the order of any product of two different translations of an STS of size n > 3 is odd, then the multiplication group Mult(L ) of the corresponding totally symmetric Steiner loop L given in Remark 3 of order n + 1 contains the alternating group of order n + 1. For a simple Steiner loop of affine type of order n to obtain that the multiplication group Mult(L S ) contains the group A n it is enough to prove that the group Mult(L S ) contains one of the permutations σ x , with x ∈ L S . Proof. If one has σ x ∈ Mult(L S ), then the permutation σ −1 x λ x = (Ω, x, −x) is a 3-cycle in the primitive subgroup Mult(L S ) of A n . By Jordan's theorem on permutations (see also Exercise 5.6.2 in [4]), Mult(L S ) = A n . This proves assertion (i).
(ii) If the order of the permutation σ x is 3k + 2 for some x ∈ L S , then one has σ x = λ 3k+3 x . If the order of the permutation σ x is 3k + 1 for some x ∈ L S , then one has σ x = λ 3(2k+1) x . Hence σ x ∈ Mult(L S ) and we are done by part (i).

Remark 12.
If n = 9, then the group Mult(L S ) is the elementary abelian 3-group with two generators, and σ x is the product of two 3-cycles (see Theorem 6). In this case, a direct computation shows that the group Σ is the alternating group A 8 .
Consider now the Steiner loop L S of affine type corresponding to the Steiner triple system on 13 points S 2 defined in 2. Proof. The claim follows from Jordan's theorem on the symmetric group S n , because, by Theorem 4, λ 2 x is a 3-cycle in the primitive group Mult(L S ). Corollary 4. If R ≤ S is a Steiner triple subsystem of order 7, then the subloop L R of L S has the alternating group A 7 as its multiplication group. Proof. We denote by S 1 and S 2 the two non-isomorphic STS(13) as they are defined in [23], p. 152-153. Taking the left translation λ 1 = (0, 1, 4)(2, 7, 3, 10, 6) (5,8,11,9,12) ∈ S 1 and the left translation λ 2 = (0, 1, 4)(2, 12, 5, 10, 6) (3,8,11,9,7) ∈ S 2 , in both cases, the fifth power of such an element is a 3-cycle. Since Mult(L S ) is a primitive subgroup of A 13 , then Mult(L S ) = A 13 by Jordan's theorem. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.