LOOPS ’11, OPEN PROBLEM SESSION
The following problems and conjectures were presented during the Loops ’11
Open Problem Session, Třešt’, Czech Republic, July 21–27, 2011.
1. Moufang Loops
Conjecture 1.1 (A. Grishkov). Let L be a finite Moufang loop without normal
abelian subgroups. Then there exists a normal subloop L0 E L such that
(i) L/L0 ' C2n is an elementary abelian 2-group.
(ii) L0 ' M × G, where G is a group without normal abelian subgroups, and M
is a direct product of simple non-associative subloops.
Conjecture 1.2 (A. Grishkov). Let F = hx, y, zi be a free Moufang loop with free
generators x, y, z. Then
(i) F is torsion-free,
(ii) F i /F i+1 is a torsion-free abelian group, where F i is the normal subloop
of F generated by all words of commutator-associator length ≥ i. Here,
x, y, z are words of length 1 and if v, w, u are some words of length i, j, k,
respectively, then [v, w] and (v, w, u) are words of length i + j and i + j + k,
respectively.
(iii) ∩i F i = 1.
Conjecture 1.3 (A. Grishkov). Let M = ha, b, ci be the free Malcev algebra over
the
P∞rational field Q with free generators a, b, c. Consider the algebra M̃ of series
i=1 ai , ai ∈ M , degai = i. Let L = (M̃ , ?) be the Moufang loop with the CampbellHausdorff multiplication
[x, y]
+ ...
x?y =x+y+
2
Then the subloop of L generated by a, b, c is a free 3-generated Moufang loop.
Conjecture 1.4 (A. Grishkov). Conjectures 1.2 and 1.3 are equivalent.
Conjecture 1.5 (A. Grishkov, The Restricted Burnside Problem for Moufang
Loops). Let M be a finite Moufang loop of exponent n with m generators. Then
there exists a function f (n; m) such that |M | < f (n; m).
[Notes: In the case n = p 6= 3 is prime, the conjecture was proved by Grishkov.
If p = 3 and M is commutative, it was proved by Bruck. The general case for p = 3
was proved by G. Nagy. The case n = pm holds by the Grishkov-Zelmanov Theorem
(preprint).]
Conjecture 1.6 (A. Grishkov, The Sanov and M. Hall Theorems for Moufang
Loops). Let L be a finitely generated Moufang loop of exponent 4 or 6. Then L is
finite.
Conjecture 1.7 (A. Grishkov, Analog of Hall-Higman Theorem for Moufang
Loops). Let G be a finite m-generated solvable group with triality,
S3 = {σ, ρ | σ 2 = ρ3 = (σρ)2 = 1} ≤ Aut(G).
1
2
LOOPS ’11, OPEN PROBLEM SESSION
Suppose that G = [G, S3 ], M (G) = [G, σ], and xn = 1 for every x ∈ M (G).
Then there exists a function f (m, n, p), where p is prime, such that the p-length
of G is less than f (m, n, p).
Conjecture 1.8 (A. Grishkov). Conjectures 1.5 and 1.7 are equivalent.
Conjecture 1.9 (A. Grishkov). Let fk = [. . . [[x1 , x2 ], x3 ] . . . xk ] and let Mk,m be
a Moufang loop with m generators which satisfies the identity fk = 1. Then Mk,m
is nilpotent.
[Notes: For k = 2, this is the famous Bruck Theorem.]
Conjecture 1.10 (A. Grishkov). Let M (k) be a solvable algebraic connected Moufang loop over an algebraically closed field of characteristic p > 0. Then M (k) =
T.N , where T is an algebraic torus, N is a nilpotent maximal subloop.
Conjecture 1.11 (A. Grishkov). Let L be a finite Moufang loop and let L2 be the
normal subloop generated by {x2 | x ∈ L}. Suppose that Nuc(L2 ) = 1. Denote by
R the maximal solvable subloop of L2 . Then we have one of three possibilities:
(i) L2 is a direct product of two subloops,
(ii) There exists a commutative associative ring K such that L2 /R is a subgroup
of P SL2 (K). Moreover, L2 contains a subgroup L1 ' L2 /R. If R =
R1 > R2 > · · · > Rn > 1 is the low central series of R then R/R2 is a
direct sum of L1 -modules of type V ⊆ VK , where VK is 2-dimensional nonassociative P SL2 (K)-module. Moreover, Ri /Ri+1 is a direct sum of two
types L1 -modules V and W , where V is as above and W is the 3-dimensional
standard associative L1 -module.
(iii) There exists a commutative associative ring K such that L2 /R is a subgroup
of P SSL(K)−loop of elements with norm 1 in Cayley-Dicson alternative
algebra O(K). Moreover, L2 contains a subgroup L1 ' L2 /R. If R = R1 >
R2 > · · · > Rn > 1 is the low central series of R then Ri /Ri+1 is a direct
sum of L1 −modules of type V ⊆ VK , where VK is 7-dimensional (unique
with properties!!) P SL2 (K)−module.
T
Conjecture 1.12 (A. Grishkov). Let L be a finite Moufang loop, Φ(L) = {L0 ∈
M ax(L)}, where M ax(L) is the set of all proper maximal subloops of L. Then
Φ(L) is a normal nilpotent subloop.
Conjecture 1.13 (A. Grishkov). Let L be a Moufang loop such that the corresponding minimal group with triality is metabelian. Then L is a group.
Problem 1.14 (A. Grishkov). Let F be the free 3-generated Moufang loop with the
free generators a, b, c and let v = (a2 b.c)(a.bcb), w = (a2 b.ca).bcb be elements of F .
(a) Does the identity v = w hold for all Moufang loops?
(b) If v = w is not an identity, then find the maximal k such that vw−1 ∈ F k
(see Conjecture 1.2). [Note: If there is no such k, then vw−1 ∈ ∩i F i and
Conjecture 1.2(iii) is false.]
Problem 1.15 (A. Grishkov). Let L be a Moufang loop with m generators of prime
exponent p such that any subloop of L is finite. Is L finite?
Problem 1.16 (A. Grishkov). Let SF be a free metabelian Moufang loop. Does
SF have elements of finite order?
LOOPS ’11, OPEN PROBLEM SESSION
3
Problem 1.17 (A. Grishkov). Find all Moufang identities SLM = {vi = wi | i ∈
I} of positive type (i.e. such that vi , wi are words in the generators without negative
exponents) which are not direct consequences of the diassociativity. Examples are
the left, right and central Moufang identities, or the identity (ab)c(ab) = (a(bc)a)b.
The identity v = w from Problem 1.14 is a possible candidate for SLM . Note that
the free diassociative semi-loop with the identities SLM may be embedded in a free
Moufang loop and thus may be called a free Moufang semi-loop.
Problem 1.18 (J.D. Phillips). Let L be a Moufang loop and C(L) = {x ∈ L; xy =
yx for every y ∈ L}. Find an equational proof of C(L) E L.
[Note: Gagola’s proof of C(L) E L uses groups with triality.]
Problem 1.19 (A. Rajah). We say that a variety V of loops satisfies Moufang
theorem if for every Q ∈ V the following implication holds: for every x, y, z ∈ Q,
if x(yz) = (xy)z then hx, y, zi is a group. Is every variety that satisfies Moufang
theorem contained in the variety of Moufang loops?
Problem 1.20 (Y. Movsisyan). An algebra (Q, Σ) is left M-linear if there is a
Moufang loop (Q, ◦) such that for every A ∈ Σ there is ϕ ∈ Aut(Q, ◦) and a
permutation σ of Q such that A(x, y) = ϕ(x) ◦ σ(y) for every x, y ∈ Q.
(i) Characterize the class of left M-linear algebras.
(ii) Characterize the class of algebras that are both left and right M-linear.
Conjecture 1.21 (V. Ursu). Let Q be the octonion loop of order 16. Then the
theory of quasi-identities true in Q does not have an independent basis of quasiidentities.
Problem 1.22 (W.L. Chee and A. Rajah). Are
orders associative?
(a) p4 q 3 , where 3 < p < q are primes, q 6≡ 1
(b) pqr4 , where p ≤ q < r are primes, r 6≡ 1
(c) pq 4 r, where p < q < r are primes, q 6≡ 1
all Moufang loops of the following
(mod p),
(mod p), r 6≡ 1 (mod q),
(mod p).
Problem 1.23 (W.L. Chee and A. Rajah). Classify all nonassociative loops of
order 3p3 , where p is a prime, p ≡ 1 (mod 3).
Problem 1.24 (W.L. Chee and A. Rajah). Obtain the product rules for the nonassociative Moufang loops of order p5 , p > 3.
2. Other
Problem 2.1 (A. Krapež). Solve the functional equation F (x, y, y) = x for a
ternary quasigroup F .
[Notes: Three (families of ) solutions readily come to mind.
(a) Let · be a binary quasigroup and / its right division. One solution of
the equation is F (x, y, z) = xy/z but there are many similar ones like
F (x, y, z) = y/(x\z).
(b) Let · be a (α, β, γ)–inverse quasigroup (α(xy) · β(x) = γ(y)) and define
F (x, y, z) = γ −1 (α(yx) · β(z)). In this case we can also use other types of
inverse quasigroups to obtain similar solutions of the equation.
(c) Let · be any quasigroup and B a unipotent quasigroup (B(x, x) = b for some
element b). Define %(x) = xb and F (x, y, z) = %−1 (x · B(y, z)).]
4
LOOPS ’11, OPEN PROBLEM SESSION
Problem 2.2 (T. Kepka on behalf of J. Ježek). An algebra is uniserial if its subloop
lattice Sub(A) is a chain. Is there a loop L with Sub(L) isomorphic to one of: R,
Q, Z, an infinite reversed down ordinal?
Problem 2.3 (Y. Movsisyan). Characterize the class of multiplicative quasigroups
of quasifields.
Problem 2.4 (Y. Movsisyan). Investigate the number of pairs of orthogonal subsquares of a latin square.
Problem 2.5 (F. Sokhatsky). Let Φ be a set of multiary operations. Then (Φ, ∗, σ, τ )
is a bi-unary semigroup of operations if
(f ∗ g)(x0 , . . . , xp ) = f (g(x0 , . . . , xn ), xn+1 , . . . , xp ),
(σg)(x0 , . . . , xn ) = g(xn , x0 , . . . , xn−1 ),
(τ g)(x0 , . . . , xn ) = g(x1 , x0 , x2 , . . . , xn ).
(i) Find an abstract characterization of the class of all bi-unary semigroups of
operations.
(ii) Find an abstract characterization of the class of all bi-unary semigroups of
quasigroups, i.e., when Φ is a set of multiary quasigroup operations.
Problem 2.6 (G. Nagy). Find a new algorithm for random generation of semifields
of dimension n over their center Fq .
Problem 2.7 (G. Nagy). Is there a loop Q such that Mltr (Q) ≤ P Sp(2n, q), q an
odd prime?
Problem 2.8 (I. Wanless). Fix a prime p. Is there a family of latin squares with
more than cubically many subsquares of order p? More precisely, is it true that for
every constant c there is a latin square L of order n such that there are more than
cn3 subsquares of order p in L?
[Notes: The answer is “no” for p = 2, 3, 5. For p = 7, the subsquares would
have to be multiplication tables of a Steiner quasigroup.]
Problem 2.9 (I. Wanless). Let Q be a loop in which for every x, y ∈ Q the
bijections Lx , Rx Ry−1 have order 3 or 1. Does it follow that |Lx L−1
y | ∈ {1, 3} for
every x, y ∈ Q?
Problem 2.10 (I. Wanless). A loop Q is van Rees if every loop isotope of Q has
exponent 3. Let Q be a van Rees loop, and let (Q, ∗) be defined by x ∗ y = x(y(yx)).
Is Q a van Rees loop? Is it a 3-loop (of order a power of 3)?
[Note: It is known that in this context (Q, ∗) is a Steiner loop.]
Problem 2.11 (P. Vojtěchovský). A loop Q is automorphic if every inner mapping
of Q is an automorphism of Q.
(i) Is there a finite simple nonassociative automorphic loop?
(ii) Does |S| divide |Q| for every finite automorphic loop Q and its subloop S?
(iii) Do Sylow-p and Hall-π Theorems hold for automorphic loops?
(iv) Classify automorphic loops of order p3 , p a prime.
(v) Classify commutative automorphic loops of order p4 , p a prime.
Conjecture 2.12 (D. Keedwell). A finite neofield (N, +, ·) is a loop (N, +) with
identity element 0 such that (N \{0}, ·) is a group and the distributive laws a(b+c) =
LOOPS ’11, OPEN PROBLEM SESSION
5
ab + ac, (b + c)a = ba + ca hold. A neofield is cyclic if (N \ {0}, ·) is a cyclic group,
with a generating element x, say.
A cyclic neofield of r elements is said to have property D if
(1 + xu )/(1 + xu−1 ) = (1 + xv )/(1 + xv−1 )
=⇒
u=v
for all integers u, v taken modulo r − 1.
We conjecture that neofieds with property D exist for every order r except 2 and
6.
Equivalently, we conjecture that for every integer r except 2 and 6, the residues
2, 3, . . . , r − 2 modulo r − 1 can be arranged in a sequence Pr in such a way that the
partial sums of the first one, two, . . . , r−3 are all distinct and non-zero modulo r−1
and so that, in addition, when each element of the sequence is reduced by 1, the new
sequence Pr0 has the same property. Example: for r = 10, Pr = h7, 6, 2, 5, 8, 4, 3i,
the partial sums modulo 9 are x, x + 7, x + 4, x + 6, x + 2, x + 1, x + 5, x + 8,
Pr0 = h6, 5, 1, 4, 7, 3, 2i, the partial sums modulo 9 are x, x + 6, x + 2, x + 3, x + 7,
x + 5, x + 8, x + 1.
[Notes: Every finite field is a neofield with property D. Let s be the integer such
that xs + 1 = 0. It can be shown that, in any cyclic neofield, s = 0 if the order r of
the neofield is even, and s = (r − 1)/2 if r is odd. It can be shown that existence
of a finite property D neofield of order r (so that xr−1 = 1) implies existence of a
pair of orthogonal latin squares of order r.]