arXiv:1508.03399v2 [] 21 Dec 2017

Nilpotent Steiner loops of class 2
A. Grishkov, D. Rasskazova, M. Rasskazova, I. Stuhl
Abstract
We describe Steiner loops of nilpotency class 2 and establish the
classification of finite 3-generated nilpotent Steiner loops of nilpotency
class 2.

1

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 in L.
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 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 Ass(L)
of L is the smallest normal subloop H of L such that L/H is a group. Thus,
Ass(L) is the smallest normal subloop of L containing associators (x, y, z)
for all x, 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. [1]).
A Steiner triple system S induces a multiplication on pairs of distinct
points x, y taking as the 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
2010 Mathematics Subject Classification:20N05, 05B07.
Key words and phrases: Steiner triple system, Steiner loop, nilpotency class 2

1

points are the elements of S \ {e}, and the blocks are triples {x, y, xy} where
x, y ∈ S, x 6= y.
A loop L is said to be totally symmetric if x · y = y · x and x · (x ·
y) = y for all x, y ∈ L. A totally symmetric loop of exponent 2 is called
a Steiner loop. A variety of universal algebras is called a Schreier variety
if every subalgebra of any free algebra in that variety is also free in that
variety. Steiner loops form a Schreier variety; it is precisely the variety of
all diassociative loops of exponent 2 (see in [2] p. 310). Steiner loops are
in a one-to-one correspondence with Steiner triple systems. Multiplication
groups of Steiner loops have been studied in [7]. Central extensions of Steiner
loops and quasigroups yielding algebraic structures of Steiner triple systems
with cyclic orientations on each triple have been introduced in [8, 9].
Since Steiner loops form a variety, we can deal with free objects and
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 the word:
• X ⊂ S(X)∗ ,
• wv ∈ S(X)∗ precisely if, v, w ∈ S(X)∗ , w > v and if w = w1 w2 , then
v 6= wi , (i = 1, 2).
On S(X) = S(X)∗ ∪ {∅} we define a multiplication 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.

2

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 = (((xi2 xi1 )xi3 )...xis ) of S(X). As S(X)/S1 (X) is
2

an abelian 2-group, it is isomorphic to Fn2 . Hence, to any element v ∈ Fn2
an element σ = (i, j, ...) can be related, where i, j, ... are the numbers of
coordinates having value 1 of the vector v. Therefore, {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 to 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 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 ).

(1)

Here f : V × V −→ Z is a Steiner loop cocycle, that is, a map satisfying
f (0V , v1 ) = f (v1 , v1 ) = 0Z , 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 Steiner loop 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 )

and

g(0V ) = 0Z .

for all v1 , v2 ∈ V . Let
B2 (V, Z) = δ(C 1 (V, Z))
and
H2 (V, Z) = Z 2 (V, Z)/B2 (V, Z).
Central extensions L1 and L2 are called equivalent precisely if, there is
an isomorphism φ : Lf1 −→ Lf2 such that φ(v, ∗) = (v, ∗) if v ∈ V and
φ(∗, z) = (∗, z + λ(∗)) where z, λ(∗) ∈ Z.
Any two equivalent extensions L1 and L2 are clearly isomorphic, but the
converse is not true in general (for an example see the proof of Theorem 5).
Lemma 1 Central extensions Lf1 and Lf2 corresponding to cocycles f1 and
f2 are equivalent if and only if f1 = f2 in H2 (V, Z).
Proof. The map ϕ = (ϕ1 , ϕ2 ) : Lf1 −→ Lf2 , with ϕ1 (v, z) = v and
ϕ2 (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 H2 (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 ).

3

Let {v1 , ..., vn } be a basis of VF2 ; as before, we can identify V with Pn the set of all subsets of In . The set Pn has an ordering: σ > τ if |σ| > |τ |
or |σ| = |τ |, σ = {i1 < ... < ik }, τ = {j1 < ... < jk } with i1 = j1 , ..., is =
js , is+1 > js+1 .
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 . In what follows △ stands for
the set-theoretical difference.
Lemma 2 Z 2 (V, Z) = Z02 (V, Z) ⊕ B2 (V, Z).
Proof. First, consider the case when f ∈ Z02 (V, Z) ∩ B2 (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,
X

f (σ, τ ) = g(σ△τ ) + g(σ) + g(τ ) =

g({i}) +

X

g({i}) +

X

g({i}) +

g({i}) +

X

g({i}) +

i∈τ ∩σ

i∈σ\τ

X

X

g({i})

i∈τ

g({i})

i∈σ∩τ

i∈τ \σ

i∈σ\τ

+

X

g({i}) +

i∈σ

i∈σ△τ

=

X

X

g({i}) = 0.

i∈τ \σ

Now, suppose f P
∈ Z 2 (V, Z). For σ = (i1 , ..., ik ) we define σ s = (i1 , ..., is−1 ),
s > 1, and g(σ) = ks=2 f (σ s , {is }), assuming that |σ| > 1 and g({i}) = 0.
Then f + δ(g) ∈ Z02 (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

s

f (σ , {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) ∈ Z02 (V, Z) completing the proof of the lemma.
We call a pair (σ, τ ) regular if and only if σ△τ > σ > τ . Note, that if
∅ =
6 σ 6= τ 6= ∅ then precisely one of the pairs (σ, σ△τ ), (σ, τ ), (σ△τ, τ ),
(σ△τ, σ), (τ, σ), (τ, σ△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} and
|σ| > 1.
4

Lemma 3 The number of elements of the set of all strongly regular pairs is
1 2n−1
(2
+ 1) − 3 · 2n−1 + n + 1.
3
Proof. Let P be the set of all ordered pairs (σ, τ ), σ > τ 6= ∅. Then
2
2
|P | = Cm
, m = 2n − 1 and hence, the number of regular pairs is 13 Cm
=
1 n
n−1
(2
−
1)(2
−
1).
3
If (σ, τ ) is regular but not strongly regular then τ = {i}, i ≥ max(σ) or
σ = {j}, j > i. Hence, for given i we have (2i−1 + n − 2i) regular but not
strongly regular pairs. Then the number of strongly regular pairs equals
n

X
1 n
(2 − 1)(2n−1 − 1) −
(2i−1 + n − 2i)
3
i=2
1
= (2n − 1)(2n−1 − 1) − 2n − n2 + n(n + 1) + 1
3
1
= (22n−1 + 1) − 3 · 2n−1 + n + 1.
3

Theorem 4 The union of sets

({i}, σ \ {i}, τ ) (σ, τ ) or (τ, σ) strongly regular,
σ ∩ τ = ∅, {i} = max(σ ∪ τ ) ∈ σ
and

({j}, µ, λ) (µ, λ) strongly regular, µ ∩ λ 6= ∅, {j} = max(µ ∩ λ) .
is a basis of the 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. Let F (X) be a free 2-step nilpotent Steiner loop with free generators X = {x1 , ..., xn }. F (X) can be realized as a central extension Lf on
V × Z for some Steiner loop cocycle f ∈ Z02 (V, Z).

5

For any elements (σ, s), (µ, m), (τ, t) ∈ Lf we have
 

 

(σ, s), (µ, m), (τ, t) = (σ, s) ◦ (µ, m) ◦ (τ, t) ◦ (σ, s) ◦ (µ, m) ◦ (τ, t)

 

= (σ△µ, f (σ, µ) + s + m) ◦ (τ, t) ◦ (σ, s) ◦ (µ△τ, f (µ, τ ) + m + t)
= (σ△µ△τ, f (σ, µ) + f (σ△µ, τ ) + s + m + t)
◦(σ△µ△τ, f (µ, τ ) + f (σ, µ△τ ) + s + m + t)
= (σ△µ△τ △σ△µ△τ, f (σ, µ) + f (σ△µ, τ ) + f (µ, τ ) + f (σ, µ△τ ))
= (∅, f (σ, µ) + f (σ△µ, τ ) + f (µ, τ ) + f (σ, µ△τ )).
Taking s = m = t = 0 and identifying (σ, 0), (µ, 0) and (τ, 0) with σ, µ and
τ , respectively, we obtain the following relation involving associators:
(σ, µ, τ ) = f (σ, µ) + f (µ, τ ) + f (σ△µ, τ ) + f (σ, µ△τ ).

(3)

Set Zf = f (V, V ). Then by (3) we get that Ass(F (X)) ⊆ Zf .
Next, we show that Zf ⊆ Ass(F (X)). Let σ, τ ∈ V be such that σ > τ . If
the pair (σ, τ ) is not regular, then σ > σ△τ and f (σ, τ ) = f (σ△τ, σ) by the
properties of Steiner loop cocycles. Note that the pair (σ△τ, σ) is already
regular. Now, suppose that (σ, τ ) is regular but not strongly regular. Then
τ = {i} and {i} ≥ max(σ) or σ = {j}, τ = {i} and j > i. In this case
f (σ, {i}) = 0 by the definition of Z02 (V, Z). This means that it is enough to
show that f (σ, τ ) ∈ Ass(F (X)) for any strongly regular pair (σ, τ ).
Let (σ, τ ) be a strongly regular pair, that is, |σ| ≥ |τ | > 1 or |σ| ≥ |τ | = 1
but {i} < max(σ), τ = {i} and |σ| > 1. Furthermore, let {i} = max(σ ∪ τ ).
In what follows we use induction in r := |σ| + |τ |. Assume that f (σ, τ ) ∈
Ass(F (X)) if (σ, τ ) is strongly regular and |σ| + |τ | < r. Consider the case
where the pair (σ, τ ) is strongly regular and |σ| + |τ | = r.
If σ ∩ τ = ∅ and {i} ∈ σ, then by (3) we have:
({i}, σ \ {i}, τ ) = f ({i}, σ \ {i}) + f (σ \ {i}, τ )
+f ({i}△(σ \ {i}), τ ) + f ({i}, (σ \ {i})△τ )
and
f (σ, τ ) = ({i}, σ \ {i}, τ ) + f ({i}, σ \ {i}) + f (σ \ {i}, τ )
(4)
+f ({i}, (σ \ {i})△τ ) = ({i}, σ \ {i}, τ ) + f (σ \ {i}, τ ),
6

since f ({i}, σ \ {i}) = 0 and f ({i}, (σ \ {i})△τ ) = 0. By the induction
assumption, f (σ \ {i}, τ ) ∈ Ass(F (X)) as |σ \ {i}| + |τ | < |σ| + |τ | and hence
f (σ, τ ) ∈ Ass(F (X)). Similarly, we can prove the same fact in the case where
σ ∩ τ = ∅ and {i} ∈ τ .
If σ ∩ τ 6= ∅ and {j} = max(σ ∩ τ ), then by (3) we obtain that
({j}, σ, τ ) = f ({j}, σ) + f (σ, τ ) + f ({j}△σ, τ ) + f ({j}, σ△τ )
and
f (σ, τ ) = ({j}, σ, τ ) + f ({j}, σ) + f (σ \ {j}, τ ) + f ({j}, σ△τ ).

(5)

Each summand in the right-hand side is contained in Ass(F (X)). Namely, as
f is a Steiner loop cocycle, we have f ({j}, σ) = f (σ, {j}) = f (σ \ {j}, {j}).
Then |σ \ {j}| + |{j}| = |σ| < |σ| + |τ | yields by the induction hypothesis that
f ({j}, σ) ∈ Ass(F (X)). Similarly, we have that f (σ \ {j}, τ ) ∈ Ass(F (X)) by
induction, since {j} ∈ σ and |σ \ {j}| + |τ | < |σ| + |τ |. Also, f ({j}, σ△τ ) ∈
Ass(F (X)) by the induction, since |{j}| + |σ△τ | = 1 + |σ| + |τ | − 2|σ ∩ τ | <
|σ| + |τ | as σ ∩ τ 6= ∅. This implies that f (σ, τ ) ∈ Ass(F (X)).
Summarizing the above discussions, we get that Zf ⊆ Ass(F (X)). Hence
Zf = f (V, V ) = Ass(F (X)) = S1 (X)/S2 (X).
Moreover, by (4) and (5) we obtain that
n
f (V, V ) ⊆ SpanF2 ({i}, σ \ {i}, τ ), ({j}, µ, λ)
(σ, τ ), (µ, λ) strongly regular, σ ∩ τ = ∅, {i} = max(σ ∪ τ )
o

and µ ∩ λ 6= ∅, {j} = max(µ ∩ λ) =: W.
Finally, we have that f (V, V ) ⊆ W ⊆ Ass(F (X)) = f (V, V ) which implies
Ass(F (X)) = W and
dim(S1 (X)/S2 (X)) = dim(f (V, V )) = dim(Ass(F (X))) = dim(W )
= {strongly regular pairs} = 31 (22n−1 + 1) − 3 · 2n−1 + n + 1
where the last equality holds by Lemma 3.
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 id SteinerLoop(16, 2); the label 2 indicates
7

the system order as in the list established in monograph [1]. The Steiner
loop S16 is 3-generated and has the nilpotency class 2. In what follows we
describe all 3-generated Steiner loops of nilpotency class 2.
Theorem 5 There exist three non-isomorphic non-associative 3-generated
Steiner loops of nilpotency class 2 and their orders are 16, 32 and 64, respectively.
Proof. Let S(X = {x1 , x2 , x3 }) be the 3-generated free Steiner loop of
nilpotency class 2 and Z =< (z1 , z2 , z3 ) > be a center of S(X). By Theorem
4, we can choose z1 = (x1 , x2 , x3 ), z2 = (x2 , x1 , x3 ), z3 = (x3 , x2 , x1 x3 ).
Let G = AutS(X) be a group of automorphisms of the loop S(X). Since
S(X) is free and Z is a G−invariant subloop of S(X), we have an epimorphism
φ : G → GL3 (F2 ) with ker(φ) = {ρ ∈ G|xρ = xz, z ∈ Z}.
The group G acts on the F2 −space Z, and this action depends only on
the images of the elements of G in GL3 (F2 ). As the group GL3 (F2 ) is simple
and has no non-trivial 2−dimensional F2 −representations, the G−module Z
is irreducible.
If P is a 3-generated Steiner loop of nilpotency class 2, then there is a
canonical epimorphism π : S(X) → P and ker(π) ⊆ Z. Note that for any
other 3-generated Steiner loop Q and canonical epimorphism ψ : S(X) → Q
the loops P and Q are isomorphic if and only if there exists ̺ ∈ GL3 (F2 )
such that ker(π)̺ = ker(ψ). Indeed, if ker(π)̺ = ker(ψ) then ̺ induces an
isomorphism ̺¯ : P = S(X)/ker(π) −→ S(X)/ker(ψ) = Q. Conversely, let
̺¯ : P = S(X)/ker(π) −→ S(X)/ker(ψ) = Q be an isomorphism between P
and Q. Then ̺¯ induces a homomorphism υ : S(X) −→ S(X)/ker(ψ) with
υ = ̺¯ ◦ π. For every x ∈ X one can choose σ(x) ∈ X such that υ(x) =
σ(x)ker(ψ) ∈ S(X)/ker(ψ). Since the loop S(X) is free, there is a unique
homomorphism ῡ : S(X) −→ S(X) satisfying ῡ(x) = υ(x) for all x ∈ X. It is
easy to see that ῡ ∈ AutS(X) and ῡ(ker(π)) = ker(ψ).
The group GL3 (F2 ) acts transitively on Z \ {1} and on the set of the
two dimensional F2 −subspaces of Z. As Z is a three dimensional irreducible
GL3 (F2 )−module, there exists a unique 3-generated Steiner loop of nilpotency class 2 for each of the orders 16, 32 and 64.
Acknowledgement
The first autor is grateful to FAPESP and CNPq, Brazil; to RFBR, Russia, grant 16-01-0577a (pp. 1-5) and to Russian Science Foundation, project
16-11-10002 (pp. 6-10) for financial support. The second author has been
supported by FAPESP Grant - 2015/17611-8. The third author has been
supported by CNPq, process 307824/2016-0. The fourth author has been
8

supported by FAPESP Grant - 11/51845-5, and expresses her gratitude to
IMS, University of São Paulo and the Math. Dept., Penn State University,
for the warm hospitality.
The authors are grateful to the referee for careful reading of the manuscript
and a numerous queries aiming at a substantial improvement of the paper.

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systems and their automorphism groups. Journal of Algebra and its Applications, 14, (2015).
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their properties. Ars. Combinatoria, 15, (1983), 3 – 110.
[5] G. P. Nagy, P. Vojtěchovský, LOOPS: Computing with quasigroups and
loops in GAP. available at http://www.math.du.edu/∼petr/loops
[6] H. O. Pflugfelder, Quasigroups and Loops: Introduction. Heldermann
Verlag, Berlin (1990).
[7] K. Strambach, I. Stuhl, Translation groups of Steiner loops. Discrete
Mathematics, 309, (2009), 4225 – 4227.
[8] K. Strambach, I. Stuhl, Oriented Steiner loops. Beiträge zur Algebra und
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[9] I. Stuhl, Oriented Steiner quasigroups. Journal of Algebra and its Applications, 13, (2014).
Alexander Grishkov
IMS, University of São Paulo 05508-090 São Paulo, SP, Brazil
Omsk State University n.a. F.M. Dostoevskii, Pr. Mira 55-A, 644053 Omsk,
Russia
E-mail: grishkov@ime.usp.br

9

Diana Rasskazova
IMS, University of São Paulo, 05508-090 São Paulo, SP, Brazil
E-mail: ivakirjan@gmail.com
Marina Rasskazova
Omsk State Technical University, 644050 Omsk, Russia
E-mail: marinarasskazova1@gmail.com
Izabella Stuhl
Math Dept, Penn State University, State College, PA 16802, USA
E-mail: ius68@psu.edu

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