Contributed talk at Loops’07

Formulae for loop operations

Contributed talk at Loops’07

Aleˇs Dr ´apal

Charles University, Prague

Motivation

By K. Kunen [2000] there are exactly 3 nonassociative CC loops of order p². But he gives no formula.

Motivation

By K. Kunen [2000] there are exactly 3 nonassociative CC loops of order p². But he gives no formula.

The formulae are in fact quite simple:

(a, i)(b, j) = (a + b, κab(i + j)) on _Zp × Zp, where κ = 1 or κ is a nonsquare, and a · b = a + b + pab on _Zp².

Motivation

By K. Kunen [2000] there are exactly 3 nonassociative CC loops of order p². But he gives no formula.

The formulae are in fact quite simple:

(a, i)(b, j) = (a + b, κab(i + j)) on _Zp × Zp, where κ = 1 or κ is a nonsquare, and a · b = a + b + pab on _Zp².

V. K. Belousov in his book Osnovy teorii kvazigrupp i lup constructs a family of CC loops on F × F^∗ by formula

(a, x)(b, y) = (a + x⁻¹b + (1 − x⁻¹)(1 − y⁻¹)s, xy)

Motivation

By K. Kunen [2000] there are exactly 3 nonassociative CC loops of order p². But he gives no formula.

The formulae are in fact quite simple:

(a, i)(b, j) = (a + b, κab(i + j)) on _Zp × Zp, where κ = 1 or κ is a nonsquare, and a · b = a + b + pab on _Zp².

V. K. Belousov in his book Osnovy teorii kvazigrupp i lup constructs a family of CC loops on F × F^∗ by formula

(a, x)(b, y) = (a + x⁻¹b + (1 − x⁻¹)(1 − y⁻¹)s, xy) For F = Zp all such nonassociative loop are ^∼₌ to

(a, x)(b, y) = (a + xb + (1 − x)(1 − y), xy).

Feelings and opinions

There are many results which describe a class of loops by long formulae with many parameters. There are

other results that describe loops only implicitly, by group transversals.

Feelings and opinions

There are many results which describe a class of loops by long formulae with many parameters. There are

other results that describe loops only implicitly, by group transversals.

In recent years I came across a number of results,

where the loops involved looked to be very complicated, but where after some effort a general and simple

formula was found.

Feelings and opinions

There are many results which describe a class of loops by long formulae with many parameters. There are

other results that describe loops only implicitly, by group transversals.

In recent years I came across a number of results,

where the loops involved looked to be very complicated, but where after some effort a general and simple

formula was found.

I believe more attention should be paid to exact and sufficiently short and general description of loop

operations. I also believe that eventually such an effort will open new connections.

What have these loops in common?

Define 5 classes of loops Q on _Zp×Zk, where p is a prime, 1<k<p, and s, t ∈ Z^∗_p are parameters.

What have these loops in common?

Define 5 classes of loops Q on _Zp×Zk, where p is a prime, 1<k<p, and s, t ∈ Z^∗_p are parameters.

(1) k = 2, and (a, x) · (b, y) = (a + t^x(−1)yb, x + y);

What have these loops in common?

Define 5 classes of loops Q on _Zp×Zk, where p is a prime, 1<k<p, and s, t ∈ Z^∗_p are parameters.

(1) k = 2, and (a, x) · (b, y) = (a + t^x(−1)yb, x + y);

(2) (a, x) · (b, y) = (a + bt^x + (1 − t^x)(1 − t^y), x + y);

What have these loops in common?

Define 5 classes of loops Q on _Zp×Zk, where p is a prime, 1<k<p, and s, t ∈ Z^∗_p are parameters.

(1) k = 2, and (a, x) · (b, y) = (a + t^x(−1)yb, x + y);

(2) (a, x) · (b, y) = (a + bt^x + (1 − t^x)(1 − t^y), x + y);

(3) (a, x) · (b, y) = (a + (1 − s + st^x)b, x + y);

What have these loops in common?

Define 5 classes of loops Q on _Zp×Zk, where p is a prime, 1<k<p, and s, t ∈ Z^∗_p are parameters.

(1) k = 2, and (a, x) · (b, y) = (a + t^x(−1)yb, x + y);

(2) (a, x) · (b, y) = (a + bt^x + (1 − t^x)(1 − t^y), x + y);

(3) (a, x) · (b, y) = (a + (1 − s + st^x)b, x + y);

(4) represent _Zk by a cyclic subgroup C of GL(2, p). For

x ∈ C, x = (x_ij) let λ_x = x₂₁ + x₂₂ and ρ_x = x₁₁ + x₁₂. Set (a, x) · (b, y) = (aρ_y + bλ_x, xy);

What have these loops in common?

Define 5 classes of loops Q on _Zp×Zk, where p is a prime, 1<k<p, and s, t ∈ Z^∗_p are parameters.

(1) k = 2, and (a, x) · (b, y) = (a + t^x(−1)yb, x + y);

(2) (a, x) · (b, y) = (a + bt^x + (1 − t^x)(1 − t^y), x + y);

(3) (a, x) · (b, y) = (a + (1 − s + st^x)b, x + y);

(4) represent _Zk by a cyclic subgroup C of GL(2, p). For

x ∈ C, x = (x_ij) let λ_x = x₂₁ + x₂₂ and ρ_x = x₁₁ + x₁₂. Set (a, x) · (b, y) = (aρ_y + bλ_x, xy);

(5) (a, x) · (b, y) = ((a + b)/(1 + tf^x(0)f^y(0)), x + y), where f(u) = (su + 1)/(tu + 1) for every u ∈ Zp, u 6= −t⁻¹.

What have these loops in common?

Define 5 classes of loops Q on _Zp×Zk, where p is a prime, 1<k<p, and s, t ∈ Z^∗_p are parameters.

(1) k = 2, and (a, x) · (b, y) = (a + t^x(−1)yb, x + y);

(2) (a, x) · (b, y) = (a + bt^x + (1 − t^x)(1 − t^y), x + y);

(3) (a, x) · (b, y) = (a + (1 − s + st^x)b, x + y);

(4) represent _Zk by a cyclic subgroup C of GL(2, p). For

x ∈ C, x = (x_ij) let λ_x = x₂₁ + x₂₂ and ρ_x = x₁₁ + x₁₂. Set (a, x) · (b, y) = (aρ_y + bλ_x, xy);

(5) (a, x) · (b, y) = ((a + b)/(1 + tf^x(0)f^y(0)), x + y), where f(u) = (su + 1)/(tu + 1) for every u ∈ Zp, u 6= −t⁻¹. The common feature: Inn(Q) embeds into _{Zp ⋉ Z}^∗_p.

Loops with | Inn(Q)| = pq

Let p and q be primes, 1<q<p, and let Q be a loop such that the inner mapping group Inn(Q) is of order pq.

Loops with | Inn(Q)| = pq

Let p and q be primes, 1<q<p, and let Q be a loop such that the inner mapping group Inn(Q) is of order pq.

By Kepka and Niemenmaa, Inn(Q) cannot be cyclic.

Thus q divides p − 1.

Loops with | Inn(Q)| = pq

Let p and q be primes, 1<q<p, and let Q be a loop such that the inner mapping group Inn(Q) is of order pq.

By Kepka and Niemenmaa, Inn(Q) cannot be cyclic.

Thus q divides p − 1.

If | Inn(Q)| = pq, then | Inn(Q/Z(Q))| = pq, and

Z(Q/Z(Q)) = 1. Hence every loop L with | Inn(L)| = pq is a central extension of a loop Q with | Inn(Q)| = pq and Z(Q) = 1. My goal is to characterize all such loops.

Loops with | Inn(Q)| = pq

Let p and q be primes, 1<q<p, and let Q be a loop such that the inner mapping group Inn(Q) is of order pq.

By Kepka and Niemenmaa, Inn(Q) cannot be cyclic.

Thus q divides p − 1.

If | Inn(Q)| = pq, then | Inn(Q/Z(Q))| = pq, and

Z(Q/Z(Q)) = 1. Hence every loop L with | Inn(L)| = pq is a central extension of a loop Q with | Inn(Q)| = pq and Z(Q) = 1. My goal is to characterize all such loops.

In all classes (1)-(5) there exist loops with | Inn(Q)| = pq. Can we obtain every loop Q, Z(Q) = 1, | Inn(Q)| = pq

from one these classes?

Loops with | Inn(Q)| = pq

Let p and q be primes, 1<q<p, and let Q be a loop such that the inner mapping group Inn(Q) is of order pq.

By Kepka and Niemenmaa, Inn(Q) cannot be cyclic.

Thus q divides p − 1.

If | Inn(Q)| = pq, then | Inn(Q/Z(Q))| = pq, and

Z(Q/Z(Q)) = 1. Hence every loop L with | Inn(L)| = pq is a central extension of a loop Q with | Inn(Q)| = pq and Z(Q) = 1. My goal is to characterize all such loops.

In all classes (1)-(5) there exist loops with | Inn(Q)| = pq. Can we obtain every loop Q, Z(Q) = 1, | Inn(Q)| = pq

from one these classes?

The additional class

Define a loop on _Zp × Zk by means of a fractional mapping f(u) = (ru + s)/(tu + 1):

(a, x) · (b, y) =

a + b + xf^y(0)

1 + ts⁻¹f^x(0)f^y(0), x + y

.

The additional class

Define a loop on _Zp × Zk by means of a fractional mapping f(u) = (ru + s)/(tu + 1):

(a, x) · (b, y) =

a + b + xf^y(0)

1 + ts⁻¹f^x(0)f^y(0), x + y

.

Like in other classes we do not get a loop for any choice of f automatically, but we have to choose f in such a way that any division by zero is avoided. This means, among others, that the cycle of f has to be of a relatively short length. The existence of a loop thus depends on the existence of a

rather random number-theoretical configuration.

Loop properties

(1) (a, x) · (b, y) = (a + t^x(−1)yb, x + y) and |Q| = 2p. Then N_λ = N_ρ is of index two and | Inn(Q)| = p|t|_p.

Loop properties

(1) (a, x) · (b, y) = (a + t^x(−1)yb, x + y) and |Q| = 2p. Then N_λ = N_ρ is of index two and | Inn(Q)| = p|t|_p.

(2) (a, x) · (b, y) = (a + bt^x + (1 − t^x)(1 − t^y), x + y), where |t|_p divides k, is a CC loop. For k = q the only proper CC

loop of order pq. If k = q, then | Inn(Q)| = pq.

Loop properties

(1) (a, x) · (b, y) = (a + t^x(−1)yb, x + y) and |Q| = 2p. Then N_λ = N_ρ is of index two and | Inn(Q)| = p|t|_p.

(2) (a, x) · (b, y) = (a + bt^x + (1 − t^x)(1 − t^y), x + y), where |t|_p divides k, is a CC loop. For k = q the only proper CC

loop of order pq. If k = q, then | Inn(Q)| = pq.

(3) (a, x) · (b, y) = (a + (1 − s + st^x)b, x + y): LCC loops with N_λ = N_µ ⊳ Q of order p, Q = N_λN_ρ, k = |t|_p divides p − 1,

| Inn(Q)| = pm, m the order of h−s + stⁱ; i ∈ Zi ≤ F^∗_p.

Loop properties

(1) (a, x) · (b, y) = (a + t^x(−1)yb, x + y) and |Q| = 2p. Then N_λ = N_ρ is of index two and | Inn(Q)| = p|t|_p.

(2) (a, x) · (b, y) = (a + bt^x + (1 − t^x)(1 − t^y), x + y), where |t|_p divides k, is a CC loop. For k = q the only proper CC

loop of order pq. If k = q, then | Inn(Q)| = pq.

(3) (a, x) · (b, y) = (a + (1 − s + st^x)b, x + y): LCC loops with N_λ = N_µ ⊳ Q of order p, Q = N_λN_ρ, k = |t|_p divides p − 1,

| Inn(Q)| = pm, m the order of h−s + stⁱ; i ∈ Zi ≤ F^∗_p.

(4) (a, x) · (b, y) = (aρ_y + bλ_x, xy), where x, y ∈ C ≤ GL(2, p). A-loops with N_µ ⊳ Q of order p and N_λ = N_ρ = 1.

Loop properties

(1) (a, x) · (b, y) = (a + t^x(−1)yb, x + y) and |Q| = 2p. Then N_λ = N_ρ is of index two and | Inn(Q)| = p|t|_p.

(2) (a, x) · (b, y) = (a + bt^x + (1 − t^x)(1 − t^y), x + y), where |t|_p divides k, is a CC loop. For k = q the only proper CC

loop of order pq. If k = q, then | Inn(Q)| = pq.

(3) (a, x) · (b, y) = (a + (1 − s + st^x)b, x + y): LCC loops with N_λ = N_µ ⊳ Q of order p, Q = N_λN_ρ, k = |t|_p divides p − 1,

| Inn(Q)| = pm, m the order of h−s + stⁱ; i ∈ Zi ≤ F^∗_p.

(4) (a, x) · (b, y) = (aρ_y + bλ_x, xy), where x, y ∈ C ≤ GL(2, p). A-loops with N_µ ⊳ Q of order p and N_λ = N_ρ = 1.

(5) (a, x) · (b, y) = ((a + b)/(1 + tf^x(0)f^y(0)), x + y), where

f(u) = (su + 1)/(tu+ 1). Commutative, N_µ ⊳Q of order p.

The existence

Let q < p be primes. When can we find a loop Q of order kp with | Inn(Q)| = pq, Z(Q) = 1 that belongs to one of categories (1)-(5)? Assume q|(p − 1).

The existence

Let q < p be primes. When can we find a loop Q of order kp with | Inn(Q)| = pq, Z(Q) = 1 that belongs to one of categories (1)-(5)? Assume q|(p − 1).

(1) Loops (a, x) · (b, y) = (a + t^x(−1)yb, x + y), where |t|_p = q and k = 2, always exist. Loops with parameters t and t^′ are ^∼₌ iff t = t^′. Isotopes yield 2 ^∼₌ classes (t and t⁻¹).

The existence

Let q < p be primes. When can we find a loop Q of order kp with | Inn(Q)| = pq, Z(Q) = 1 that belongs to one of categories (1)-(5)? Assume q|(p − 1).

(1) Loops (a, x) · (b, y) = (a + t^x(−1)yb, x + y), where |t|_p = q and k = 2, always exist. Loops with parameters t and t^′ are ^∼₌ iff t = t^′. Isotopes yield 2 ^∼₌ classes (t and t⁻¹).

(3) Loops (a, x) · (b, y) = (a + (1 − s + st^x)b, x + y) exist, when every |1 − s + stⁱ|_p always divides q and k = |t|_p. For k = 2 one can always choose a parameter s. For k ≥ 3 there is no clear pattern. An example: p = 43,

k = 3, q = 7, t = 6, s = 35. Isomorphism classes depend only on s. If s 6= s^′, then the loops are not isomorphic.

The class of A-loops

A more general definition on F × M, M ≤ GL(2, F): (A, i) · (B, j) = (AB, λ_Ai + jρ_B).

The class of A-loops

A more general definition on F × M, M ≤ GL(2, F): (A, i) · (B, j) = (AB, λ_Ai + jρ_B).

Condition of existence: λ_A 6= 0, ρ_A 6= 0 for all A ∈ M.

The class of A-loops

A more general definition on F × M, M ≤ GL(2, F): (A, i) · (B, j) = (AB, λ_Ai + jρ_B).

Condition of existence: λ_A 6= 0, ρ_A 6= 0 for all A ∈ M. Condition of being an A-loop:

ρ_AB − λ_AB = ρ_Aρ_B − λ_Aλ_B for all A, B ∈ M.

The class of A-loops

A more general definition on F × M, M ≤ GL(2, F): (A, i) · (B, j) = (AB, λ_Ai + jρ_B).

Condition of existence: λ_A 6= 0, ρ_A 6= 0 for all A ∈ M. Condition of being an A-loop:

ρ_AB − λ_AB = ρ_Aρ_B − λ_Aλ_B for all A, B ∈ M.

Under this condition, M ≤ M(k), where M(k) is for k ∈ F and k = ∞ formed by matrices

a k(b − a) k(a − b) b

!

and ^{a b}

−b a

! .

The class of A-loops

A more general definition on F × M, M ≤ GL(2, F): (A, i) · (B, j) = (AB, λ_Ai + jρ_B).

Condition of existence: λ_A 6= 0, ρ_A 6= 0 for all A ∈ M. Condition of being an A-loop:

ρ_AB − λ_AB = ρ_Aρ_B − λ_Aλ_B for all A, B ∈ M.

Under this condition, M ≤ M(k), where M(k) is for k ∈ F and k = ∞ formed by matrices

a k(b − a) k(a − b) b

!

and ^{a b}

−b a

! .

Then Inn(Q) = F ⋉ D ≤ F ⋉ F^∗, where D ≤ F^∗ is generated by all λ_Aρ⁻¹, λ⁻¹ λ_Aλ_B.

Examples of A-loops

In loops (A, i)(B, j) = (AB, λ_Ai + jρ_B) the center may be nontrivial. It is formed by all scalar matrices in M.

Examples of A-loops

In loops (A, i)(B, j) = (AB, λ_Ai + jρ_B) the center may be nontrivial. It is formed by all scalar matrices in M.

The simplest example for M is 1 0

0 1

!

, −1 0

0 −1

!

, 0 1

−1 0

!

, 0 −1

1 0

! . That gives a loop of order 4p when p = |F|. The factor Q over the centre is of order 2p and Inn(Q) ∼= D_2p.

Examples of A-loops

In loops (A, i)(B, j) = (AB, λ_Ai + jρ_B) the center may be nontrivial. It is formed by all scalar matrices in M.

The simplest example for M is 1 0

0 1

!

, −1 0

0 −1

!

, 0 1

−1 0

!

, 0 −1

1 0

! . That gives a loop of order 4p when p = |F|. The factor Q over the centre is of order 2p and Inn(Q) ∼= D_2p.

For p = 23 the following matrices give Q,

|Q| = 69 = 23 · 3, where | Inn(Q)| = 253 = 23 · 11:

Commutative loops

Let (a, i)(b, j) =

a+b

1+tfⁱ(0)f^j(0), i + j

, where f(x) = ^sx+1_tx+1. The simplest case is when s = −1.

Commutative loops

Let (a, i)(b, j) =

a+b

1+tfⁱ(0)f^j(0), i + j

, where f(x) = ^sx+1_tx+1. The simplest case is when s = −1.

Then (a, i)(b, j) = (h^ij(a + b), i + j) is an operation on F × {0, 1}, h = (1 + t)⁻¹, with inner mapping group ^∼₌ to F ⋉ hhi ≤ F ⋉ F^∗.

Commutative loops

Let (a, i)(b, j) =

a+b

1+tfⁱ(0)f^j(0), i + j

, where f(x) = ^sx+1_tx+1. The simplest case is when s = −1.

Then (a, i)(b, j) = (h^ij(a + b), i + j) is an operation on F × {0, 1}, h = (1 + t)⁻¹, with inner mapping group ^∼₌ to F ⋉ hhi ≤ F ⋉ F^∗.

Examples with |Q| = pk and | Inn(Q)| = pq are for k ≥ 3 rarer. For (p, k, q) = (11, 3, 5) one can use (s, t) = (5, 2).

Connections to group cohomologies

γ(i, j) = ^c_bfⁱ(0)f^j(0) + 1 is a commutative cocycle

Zk × Zk → F^∗ when f(x) = ^ax+b_cx+d, k is the length of the cycle of f, and fⁱ(0)f^j(0) 6= −^b_c for all i, j.

Connections to group cohomologies

γ(i, j) = ^c_bfⁱ(0)f^j(0) + 1 is a commutative cocycle

Zk × Zk → F^∗ when f(x) = ^ax+b_cx+d, k is the length of the cycle of f, and fⁱ(0)f^j(0) 6= −^b_c for all i, j.

This is a connection with 2-cohomologies. Connections to 1-cohomologies are provided by CC-loops. A

1-cocycle satisfies γ(g)^hγ(h) = γ(gh). An associator of a CC loop Q is dependent only upon classes modulo N. It can be regarded as a mapping (Q/N)³ → A(Q).

Connections to group cohomologies

γ(i, j) = ^c_bfⁱ(0)f^j(0) + 1 is a commutative cocycle

Zk × Zk → F^∗ when f(x) = ^ax+b_cx+d, k is the length of the cycle of f, and fⁱ(0)f^j(0) 6= −^b_c for all i, j.

This is a connection with 2-cohomologies. Connections to 1-cohomologies are provided by CC-loops. A

1-cocycle satisfies γ(g)^hγ(h) = γ(gh). An associator of a CC loop Q is dependent only upon classes modulo N. It can be regarded as a mapping (Q/N)³ → A(Q).

The associators satisfy

[xy, u, v] = [x, u, v]^y[y, u, v], [u, xy, v] = [u, x, v]^y[u, y, v],