arXiv:1508.03399v1 [math.GR] 14 Aug 2015
Steiner loops of nilpotency class 2 A. Grishkov, D. Rasskazova, M. Rasskazova, I. Stuhl Abstract We describe Steiner loops of nilpotency class two.
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Introduction and Preliminaries
A loop is a set L with a binary operation · and a neutral element 1 ∈ L, such that for every a, b ∈ L the equations a · x = b and y · a = b have unique solutions. The center of a loop L is an associative subloop Z(L) consisting of all elements of L which commute and associate with all other elements of L. A loop L is nilpotent if the series L, L/Z(L), [L/Z(L)]/Z[L/Z(L)] ... terminates at 1 in finitely many steps. In particular, L is of nilpotency class two if L/Z(L) 6= 1 and L/Z(L) 6= 1 is an abelian group. For x, y, z ∈ L, define the associator (x, y, z) of x, y, z by (xy)z = (x(yz))(x, y, z). The associator subloop A(L) of L is the smallest normal subloop H of L such that L/H is a group. Thus, A(L) is the smallest normal subloop of L containing associators (x, y, z) for allx, y, z ∈ L. A Steiner triple system is an incidence structure consisting of points and blocks such that every two distinct points are contained in precisely one block and any block has precisely three points. A finite Steiner triple system with n points exists if and only if n ≡ 1 or 3 (mod 6) (cf. [6], V.1.9 Definition). A Steiner triple system S provides a multiplication on pairs of different points x, y taking as product the third point of the block joining x and y. Defining x · x = x, we get a Steiner quasigroup associated with S. Adjoining an element e with ex = xe = x, xx = e, we obtain a Steiner loop S, see below. Conversely, a Steiner loop determines a Steiner triple system whose points are the elements of S \ e, and the blocks are triples {x, y, xy} where x, y ∈ S, x 6= y. 2010 Mathematics Subject Classification:20N05, 05B07. Key words and phrases: Steiner triple systems, Steiner loops, nilpotency index 2
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A totally symmetric loop of exponent 2 is called Steiner loop. Steiner loops form a Schreier variety; it is precisely the variety of all diassociative loops of exponent 2. Steiner loops are in a one-to-one correspondence with Steiner triple systems (see in [2] p. 310). Since Steiner loops form a variety, we can deal with free objects. Moreover, according to [3], we have the following. Let X be a finite ordered set and let W (X) be a set of non-associative X-words. The set W (X) has an order, >, such that v > w if and only if |v| > |w| or |v| = |w| > 1, v = v1 v2 , w = w1 w2 , v1 > w1 or v1 = w1 , v2 > w2 . Next, we define the set S(X)∗ ⊂ W (X) of S-words by induction upon the length of word: • X ⊂ S(X)∗ , • wv ∈ S(X)∗ precisely if, v, w ∈ S(X)∗ , |v| ≤ |w|, v 6= w and if w = w1 · w2 , then v 6= wi , (i = 1, 2). On S(X) = S(X)∗ ∪ {∅} we define a multiplication ( still denoted by ·) in the following manner: 1. v · w = w · v = vw if vw ∈ S(X), 2. (vw) · w = w · (vw) = w · (wv) = (wv) · w = v, 3. v · v = ∅. A word v(x1 , x2 , ..., xn ) is irreducible if v ∈ S(X)∗ . The set S(X) with the multiplication as above is a free Steiner loop with the set of free generators X. In what follows we discuss Steiner loops of nilpotency class 2.
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Centrally nilpotent Steiner loops of class 2
Let S(X) > S1 (X) > S2 (X) > ... be a central series of the free Steiner loop S(X) with free generators X = {x1 , ..., xn }. Then V = S(X)/S1 (X) is an F2 space of dimension n := |X|. Given σ = {i1 < i2 < ... < is } ⊆ In define the corresponding element σ = (((xi1 xi2 )xi3 )...xis ) of S(X). Hence {σ|σ ⊆ In } is a set of representatives of S(X)/S1 (X). Determine a set of representatives of Z = S1 (X)/S2 (X). Set Lf be a central extension of F2 -spaces V and Z in the variety of Steiner loops. It is well known that Lf is a central f -extension of Z by V if and only if Lf is isomorphic to a loop defined on V × Z by the multiplication (v1 , z1 ) ◦ (v2 , z2 ) = (v1 + v2 , f (v1 , v2 ) + z1 + z2 ). 2
(1)
Here f : V × V −→ Z is a cocycle, that is, a map satisfying f (0, v1 ) = 0, f (v1 , v1 ) = 0, f (v1 , v2 ) = f (v2 , v1 ), f (v1 +v2 , v2 ) = f (v1 , v2 ) (2) for all v1 , v2 ∈ V . Denote by Z 2 (V, Z) the set of all cocycles. Next, let C 1 (V, Z) be the set of all functions g : V −→ Z and δ : C 1 (V, Z) −→ Z 2 (V, Z) such that δ(g)(v1, v2 ) = g(v1 + v2 ) + g(v1 ) + g(v2 ). for all v1 , v2 ∈ V . Let B 2 (V, Z) = δ(C 1 (V, Z)) and H 2 (V, Z) = Z 2 (V, Z)/B 2 (V, Z). Lemma 1 Central extensions Lf1 and Lf2 corresponding to different cocycles f1 and f2 are isomorphic if and only if f1 = f2 in H 2 (V, Z). Proof. The map ϕ = (ϕ1 , ϕ2 ) : Lf1 −→ Lf2 , with ϕ1 (v, z) = v and ϕ1 (v, z) = z + g(v), determines an isomorphism if and only if f1 (v1 , v2 ) = f2 (v1 , v2 )+g(v1 +v2 )+g(v1)+g(v2), i.e., f1 = f2 in H 2 (V, Z). This is because ϕ((v1 , z1 ) ◦ (v2 , z2 )) = (v1 + v2 , f1 (v1 , v2 ) + z1 + z2 + g(v1 + v2 )) = (v1 + v2 , f2 (v1 , v2 ) + z1 + z2 + g(v1 ) + g(v2 )) = ϕ(v1 , z1 ) ◦ ϕ(v2 , z2 ). Let {v1 , ..., vn } be a basis of V over F2 ; as usually, we can identify V with Pn - the set of all subsets of In . Consider a subset Z02 (V, Z) ⊂ Z 2 (V, Z), where f ∈ Z02 (V, Z) if and only if f (σ, i) = 0, i > max(σ), σ ∈ Pn = V . Lemma 2 Z 2 (V, Z) = Z02 (V, Z) ⊕ B 2 (V, Z). Proof. First, consider the case when f ∈ Z02 (V, Z) ∩ B 2 (V, Z) Then f = δ(g), and for any σ ∈ Pn and i such that i > max(σ) we have f (σ, i) = g(σ ∪ i) + g(σ) + g(i) = 0. Then g(σ) =
P
i∈σ
g(i). Hence,
f (σ, τ ) = g(σ△τ ) + g(σ) + g(τ ) =
X
g(i) +
i∈σ△τ
=
X i∈σ\τ
g(i) +
X i∈τ \σ
g(i) +
X i∈σ∩τ
X
g(i) +
i∈σ\τ
3
g(i) +
X
g(i) +
i∈σ
X i∈τ ∩σ
g(i) +
X
g(i)
i∈τ
X i∈τ \σ
g(i) = 0.
2 Now, suppose f ∈ ZP (V, Z). For σ = (i1 , ..., ik ) we define σ s = (i1 , ..., is−1 ), k s s > 1, and g(σ) = s=2 f (σ , is ), assuming |σ| > 1, g(i) = 0. Then f + δ(g) ∈ Z 2 0 (V, Z). Indeed, if i = ik+1 > ik = max(σ) then
(f + δ(g))(σ, i) = f (σ, i) + g(σ ∪ i) + g(σ) + g(i) = f (σ, i) +
k+1 X
f (σ s , is ) +
s=2
k X
f (σ s , is ) = 0,
s=2
k+1
as σ = σ and ik+1 = i. This yields that f + δ(g) ∈ Z 2 0 (V, Z) completing the proof of the lemma. We call a pair (σ, τ ) regular if and only if |σ| + |τ | < |σ| + |σ△τ | and |σ| + |τ | < |τ | + |σ△τ | or |σ| + |τ | = |σ| + |σ△τ | but min(σ△τ ) < min(τ \ σ). (Here and below △ stands for the set-theoretical difference.) Note, that if ∅= 6 σ 6= τ 6= ∅ then precisely one of the pairs (σ, σ△τ ), (σ, τ ), (σ△τ, τ ) is regular. A regular pair is called strongly regular if |σ| ≥ |τ | > 1 or |σ| ≥ |τ | = 1 but i < max(σ), where τ = {i}. Lemma 3 The number of elements of the set of all non-ordered strongly regular pairs (σ, τ ) is 1 2n−1 (2 + 1) − 3 · 2n−1 + n + 1. 3 Proof. Let P be the set of all non-ordered pairs (σ, τ ). Then P = P0 ∪P1 , where P0 = {(σ, σ)|σ ⊆ In } and P1 = {(σ, τ )|σ 6= τ ⊆ In }. Then P1 = P2 ∪P3 where P2 = {(σ, ∅)|∅ = 6 σ ⊆ In }, P3 = {(σ, τ )|τ 6= σ 6= ∅ ⊆ In }. It is easy to see that pairs (σ△τ, τ ) and (σ△τ, σ) are contained in P0 ∪ P2 or in P3 if (σ, τ ) is contained in P0 ∪ P2 or in P3 , respectively. Since |P1 | = 2n−1 (2n − 1), |P2 | = 2n − 1 and P2 ∩ P3 = ∅, we get that |P3 | = (2n − 1)(2n−1 − 1). It means that the number of regular pairs equals 31 (2n − 1)(2n−1 − 1). If pair (σ, τ ) is regular but not strongly regular then τ = {i}, i > max(σ). Hence, for a given i we have exactly (2i−1 −1) regular but not strongly regular pairs. Then the number of strongly regular pairs equals X 1 n (2i−1 − 1) (2 − 1)(2n−1 − 1) − 3 i=2 n
1 = (2n − 1)(2n−1 − 1) − 2n + n + 1 3 1 = (22n−1 + 1) − 3 · 2n−1 + n + 1. 3
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Theorem 4 The union of sets {(i, σ \ i, τ )|(σ, τ ) strongly regular σ ∩ τ = ∅, i = max(σ ∪ τ )} and {(i, σ, τ )|(σ, τ ) strongly regular, i = max(σ ∩ τ )}. is a basis of F2 -space S1 (X)/S2 (X), Moreover, 1 dimF2 (S1 (X)/S2 (X)) = (22n−1 + 1) − 3 · 2n−1 + n + 1, 3 where n := |X|. Proof. We have the following relation involving associators: (σ, µ, τ ) = f (σ, µ) + f (µ, τ ) + f (σ△µ, τ ) + f (σ, µ△τ ).
(3)
Let (σ, τ ) be a strongly regular pair, with |σ| ≥ |τ |. We will use induction in r := |σ| + |τ | and set i = max(σ ∪ τ ) ∈ σ. Then by virtue of (3) we obtain (i, σ\i, τ ) = f (i, σ\i) + f (σ\i, τ ) + f (σ, τ ) + f (i, (σ\i)△τ ) = f (σ\i, τ ) + f (σ, τ ). Furthermore, by induction set P0 = {(i, σ\i, τ )|(σ, τ ) − strongly regular, σ ∩ τ = ∅} forms a basis of Q := {f (σ, τ )|(σ, τ ) − strongly regular, σ ∩ τ = ∅}. Set P1 = {(i, σ, τ )|(σ, τ ) − strongly regular, i = max(σ ∩ τ 6= ∅)} then P1 is a basis of Q. Note that there are exactly 80 non-isomorphic Steiner triple systems of order 15. Moreover, there is only one nilpotent non-associative Steiner loop S16 of order 16 (cf. [5]), and it corresponds to the system N.2 in [4] p. 19. Furthermore, S16 has the GAP code SteinerLoop(16, 2); the label 2 indicates the system order as in the list established in monograph [1]. Take SN (X) where X = {x1 , x2 , x3 } is the 3-generated free Steiner loop of nilpotency class 2 and let Z = h(z1 , z2 , z3 )i be a center of SN (X). Then S16 = SN (X)/Z0 , where Z0 ⊂ Z = h(z1 , z2 , z3 )i is an elementary abelian 2-group of order 4. Moreover, Aut(SN (X)) = GL3 (F2 ) · Z where Z = hϕ : xi −→ xi ti i, ti ∈ Z, (i = 1, 2, 3). Since GL3 (F2 ) acts transitively on the set of all 2-dimensional F2 −subspaces of Z ≃ F32 , there exists a unique factor loop SN (X)/Z0, where Z0 is an elementary abelian 2-group of order 4. This fact confirms that there exists a unique nilpotent non-associative Steiner loop of order 16. 5
Acknowledgement A. Grishkov is supported by FAPESP, CNPQ (Brazil) and RFFI Grants 13-01-00239, 2014/319-258 (Russia) and expresses his gratitude. I. Stuhl is supported by FAPESP Grant - process number 11/51845-5, and expresses her gratitude to IMS, University of S˜ao Paulo, Brazil and to Math Dept, University of Denver for the warm hospitality.
References [1] Ch. J. Colbourn, A. Rosa, Triple systems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, (1999). [2] B. Ganter, U. Pf¨ uller, A remark on commutative di-associative loops, Algebra Universalis, 21, (1985), 310 – 311. [3] A. Grishkov, D. Rasskazova, M. Rasskazova, I. Stuhl, Free Steiner triple systems and their automorphism groups, Journal of Algebra and its Applications, 14, (2015) [4] R. A. Mathon, K. T. Phelps, A. Rosa, Small Steiner triple systems and their properties, Ars. Combinatoria, 15, (1983), 3 – 110. [5] G. P. Nagy, P. Vojtˇechovsk´y, Loops: Computing with quasigroups and loops in GAP, version 2.2.0, available at http://www.math.du.edu/loops [6] H. O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann Verlag, Berlin (1990). Alexander Grishkov Institute of Mathematics and Statistics University of S˜ao Paulo 05508-090 S˜ao Paulo, SP, Brazil E-mail:
[email protected] Diana Rasskazova Institute of Mathematics and Statistics University of S˜ao Paulo 05508-090 S˜ao Paulo, SP, Brazil
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E-mail:
[email protected] Marina Rasskazova Omsk Service Institute 644099 Omsk, Russia E-mail:
[email protected] Izabella Stuhl IMS, University of S˜ao Paulo 05508-090 S˜ao Paulo, SP, Brazil Math Dept, University of Denver Denver, CO 80208 USA University of Debrecen H-4010 Debrecen, Hungary E-mail:
[email protected]
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