Nilpotency in automorphic loops of prime power order
aa r X i v : . [ m a t h . G R ] A ug NILPOTENCY IN AUTOMORPHIC LOOPS OF PRIME POWER ORDER
P ˇREMYSL JEDLI ˇCKA † , MICHAEL KINYON, AND PETR VOJTˇECHOVSK ´Y Abstract.
A loop is automorphic if its inner mappings are automorphisms. Using so-called associated operations, we show that every commutative automorphic loop of oddprime power order is centrally nilpotent. Starting with suitable elements of an anisotropicplane in the vector space of 2 × p , we construct afamily of automorphic loops of order p with trivial center. Introduction
A classical result of group theory is that p -groups are (centrally) nilpotent. The analogousresult does not hold for loops.The first difficulty is with the concept of a p -loop. For a prime p , a finite group has ordera power of p if and only if each of its elements has order a power of p , so p -groups can bedefined in two equivalent ways. Not so for loops, where the order of an element might notbe well defined, and even if it is, the two natural p -loop concepts might not be equivalent.However, there exist several varieties of loops where the analogy with group theory iscomplete. For instance, a Moufang loop has order a power of p if and only if each of itselements has order a power of p , and, moreover, every Moufang p -loop is nilpotent [9, 10].We showed in [12, Thm. 7.1] that a finite commutative automorphic loop has order apower of p if and only if each of its elements has order a power of p . The same is true forautomorphic loops, by [15], provided that p is odd; the case p = 2 remains open.In this paper we study nilpotency in automorphic loops of prime power order. We prove: Theorem 1.1.
Let p be an odd prime and let Q be a finite commutative automorphic p -loop.Then Q is centrally nilpotent. Since there is a (unique) commutative automorphic loop of order 2 with trivial center, cf.[11], Theorem 1.1 is best possible in the variety of commutative automorphic loops. (Thesituation for p = 2 is indeed complicated in commutative automorphic loops. By [11, Prop.6.1], if a nonassociative finite simple commutative automorphic loop exists, it has exponenttwo. We now know that no nonassociative finite simple commutative automorphic loop oforder less than 2 exists [13].)In fact, Theorem 1.1 is best possible even in the variety of automorphic loops, becausefor every prime p we construct here a family of automorphic loops of order p with trivialcenter. Mathematics Subject Classification.
Primary: 20N05.
Key words and phrases. automorphic loop, commutative automorphic loop, A-loop, central nilpotency. † Supported by the Grant Agency of the Czech Republic, grant no. 201/07/P015. .1. Background. A loop ( Q, · ) is a set Q with a binary operation · such that (i) for each x ∈ Q , the left translation L x : Q → Q ; y yL x = xy and the right translation R x : Q → Q ; y yR x = yx are bijections, and (ii) there exists 1 ∈ Q satisfying 1 · x = x · x for all x ∈ Q .The left and right translations generate the multiplication group Mlt Q = h L x , R x | x ∈ Q i .The inner mapping group Inn Q = (Mlt Q ) is the stabilizer of 1 ∈ Q . Standard referencesfor the theory of loops are [2, 3, 20].A loop Q is automorphic (or sometimes just an A-loop ) if every inner mapping of Q is anautomorphism of Q , that is, Inn Q ≤ Aut Q .The study of automorphic loops was initiated by Bruck and Paige [4]. They obtainedmany basic results, not the least of which is that automorphic loops are power-associative ,that is, for all x and all integers m, n , x m x n = x m + n . In power-associative loops, the order of an element may be defined unambiguously.For commutative automorphic loops, there now exists a detailed structure theory [11], aswell as constructions and small order classification results [12].Informally, the center Z ( Q ) of a loop Q is the set of all elements of Q which commute andassociate with all other elements. It can be characterized as Z ( Q ) = Fix(Inn Q ), the set offixed points of the inner mapping group. (See § normal subloop of Q , that is, Z ( Q ) ϕ = Z ( Q ) for every ϕ ∈ Inn Q . Define Z ( Q ) = { } , and Z i +1 ( Q ), i ≥
0, as the preimage of Z ( Q/Z i ( Q )) under the canonicalprojection. This defines the upper central series ≤ Z ( Q ) ≤ Z ( Q ) ≤ · · · ≤ Z n ( Q ) ≤ · · · ≤ Q of Q . If for some n we have Z n − ( Q ) < Z n ( Q ) = Q then Q is said to be (centrally) nilpotentof class n .1.2. Summary.
The proof of our main result, Theorem 1.1, is based on a construction from[11]. On each commutative automorphic loop ( Q, · ) which is uniquely 2-divisible ( i.e. , thesquaring map x x · x is a permutation), there exists a second loop operation ◦ such that( Q, ◦ ) is a Bruck loop (see § Q, · ) coincide with thosein ( Q, ◦ ).Glauberman [8] showed that for each odd prime p a finite Bruck p -loop is centrally nilpo-tent. Theorem 1.1 will therefore follow immediately from this and from the following result: Theorem 1.2.
Let ( Q, · ) be a uniquely -divisible commutative automorphic loop with asso-ciated Bruck loop ( Q, ◦ ) . Then Z n ( Q, ◦ ) = Z n ( Q, · ) for every n ≥ . After reviewing preliminary results in §
2, we discuss the associated Bruck loop in § § §
5, we use elements of anisotropic planes in the vector space of 2 × GF ( p ) to obtain automorphic loops of order p with trivial center. We obtain one suchloop for p = 2 (this turns out to be the unique commutative automorphic loop of order 2 with trivial center), two such loops for p = 3, three such loops for p ≥
5, and at least one(conjecturally, three) such loop for every prime p ≥ § . Preliminaries
In a loop ( Q, · ), there are various subsets of interest: • the left nucleus N λ ( Q ) = { a ∈ Q | ax · y = a · xy, ∀ x, y ∈ Q }• the middle nucleus N µ ( Q ) = { a ∈ Q | xa · y = x · ay, ∀ x, y ∈ Q }• the right nucleus N ρ ( Q ) = { a ∈ Q | xy · a = x · ya, ∀ x, y ∈ Q }• the nucleus N ( Q ) = N λ ( Q ) ∩ N µ ( Q ) ∩ N ρ ( Q ) • the commutant C ( Q ) = { a ∈ Q | ax = xa, ∀ x ∈ Q }• the center Z ( Q ) = N ( Q ) ∩ C ( Q ) .The commutant is not necessarily a subloop, but the nuclei are. Proposition 2.1. [4]
In an automorphic loop ( Q, · ) , N λ ( Q ) = N ρ ( Q ) ≤ N µ ( Q ) . In acommutative automorphic loop ( Q, · ) , Z ( Q ) = N λ ( Q ) . We will also need the following (well known) characterization of C ( Q ) ∩ N ρ ( Q ): Lemma 2.2.
Let ( Q, · ) be a loop. Then a ∈ C ( Q ) ∩ N ρ ( Q ) if and only if L a L x = L x L a forall x ∈ Q .Proof. If a ∈ C ( Q ) ∩ N ρ ( Q ), then for all x, y ∈ Q , a · xy = xy · a = x · ya = x · ay , thatis, L a L x = L x L a . Conversely, if L a L x = L x L a holds, then applying both sides to 1 gives xa = ax , i.e. , a ∈ C ( Q ), and then xy · a = a · xy = x · ay = x · ya , i.e. , a ∈ N ρ ( Q ). (cid:3) The inner mapping group Inn Q of a loop Q has a standard set of generators L x,y = L x L y L − yx , R x,y = R x R y R − xy , T x = R x L − x , for x , y ∈ Q . The property of being an automorphic loop can therefore be expressedequationally by demanding that the permutations L x,y , R x,y , T x are homomorphisms. Inparticular, if Q is a commutative loop then Q is automorphic if and only if( uv ) L x,y = uL x,y · vL x,y for every x , y , u , v .We can conclude that (commutative) automorphic loops form a variety in the sense ofuniversal algebra, and are therefore closed under subloops, products, and homomorphicimages.We will generally compute with translations whenever possible, but it will sometimes beconvenient to work directly with the loop operations. Besides the loop multiplication, wealso have the left division operation \ : Q × Q → Q which satisfies x \ ( xy ) = x ( x \ y ) = y . The division permutations D x : Q → Q defined by yD x = y \ x are also quite useful, as isthe inversion permutation J : Q → Q defined by xJ = xD = x − in any power-associativeloop. f Q is a commutative automorphic loop then for all x , y ∈ Q we have xL y,x = x, (2.1) L y,x L x − = L x − L y,x , (2.2) yL y,x = (( xy ) \ x ) − , (2.3) L x − ,y − = L x,y , (2.4) D x = D x J D x , (2.5)where the first two equalities follow from [11, Lem. 2.3], (2.3) from [11, Lem. 2.5], (2.4)is an immediate consequence of [11, Lem. 2.7], and (2.5) is [11, Lem. 2.8]. In addition,commutative automorphic loops satisfy the automorphic inverse property ( xy ) − = x − y − and ( x \ y ) − = x − \ y − , (2.6)by [11, Lem. 2.6].Finally, as in [11], in a commutative automorphic loop ( Q, · ), it will be convenient tointroduce the permutations P x = L x L − x − = L − x − L x , where the second equality follows from [11, Lem. 2.3]. Lemma 2.3.
For all x, y in a commutative automorphic loop ( Q, · )( x − ) P xy = xy , (2.7) x · xP y = ( xy ) . (2.8) Proof.
Equation (2.7) is from [11, Lem 3.2]. Replacing x with x − and y with xy in (2.7)yields xP x − · xy = x − ( xy ) and xP x − · xy = xL x,x − P x − · xy = xL x,x − P yL x,x − . Now, for everyautomorphism ϕ of Q we have xϕP yϕ = ( yϕ ) − \ ( yϕxϕ ) = ( y − \ ( yx )) ϕ = xP y ϕ . Thus x − ( xy ) = xL x,x − P yL x,x − = xP y L x,x − . Canceling x − on both sides, we obtain (2.8). (cid:3) The associated Bruck loop
A loop ( Q, ◦ ) is said to be a (left) Bol loop if it satisfies the identity( x ◦ ( y ◦ x )) ◦ z = x ◦ ( y ◦ ( x ◦ z )) . (3.1)A Bol loop is a Bruck loop if it also satisfies the automorphic inverse property ( x ◦ y ) − = x − ◦ y − . (Bruck loops are also known as K -loops or gyrocommutative gyrogroups .)The following construction is the reason for considering Bruck loops in this paper. Let( Q, · ) be a uniquely 2-divisible commutative automorphic loop. Define a new operation ◦ on Q by x ◦ y = [ x − \ ( xy )] / = [( y ) P x ] / . By [11, Lem. 3.5], ( Q, ◦ ) is a Bruck loop, and powers in ( Q, ◦ ) coincide with powers in ( Q, · ).Since we will work with translations in both ( Q, · ) and ( Q, ◦ ), we will denote left transla-tions in ( Q, ◦ ) by L ◦ x . For instance, we can express the fact that every Bol loop ( Q, ◦ ) is leftpower alternative by ( L ◦ x ) n = L ◦ x n (3.2)for all integers n . roposition 3.1. [14, Thm. 5.10] Let ( Q, ◦ ) be a Bol loop. Then N λ ( Q, ◦ ) = N µ ( Q, ◦ ) . If,in addition, ( Q, ◦ ) is a Bruck loop, then N λ ( Q, ◦ ) = Z ( Q, ◦ ) . In the uniquely 2-divisible case, we can say more about the center.
Lemma 3.2.
Let ( Q, ◦ ) be a uniquely -divisible Bol loop. Then Z ( Q, ◦ ) = C ( Q, ◦ ) ∩ N ρ ( Q, ◦ ) .Proof. One inclusion is obvious. For the other, suppose a ∈ C ( Q, ◦ ) ∩ N ρ ( Q, ◦ ). Then for all x, y ∈ Q , ( x ◦ a ) ◦ y (3.2) = ( x ◦ ( x ◦ a )) ◦ y = ( x ◦ ( a ◦ x )) ◦ y (3.1) = x ◦ ( a ◦ ( x ◦ y )) = x ◦ ( x ◦ ( a ◦ y )) (3.2) = x ◦ ( a ◦ y ) , where we used a ∈ C ( Q, ◦ ) in the second equality and Lemma 2.2 in the fourth. Sincesquaring is a permutation, we may replace x with x to get ( x ◦ a ) ◦ y = x ◦ ( a ◦ y ) for all x, y ∈ Q . Thus a ∈ N µ ( Q, ◦ ) = N λ ( Q, ◦ ) (Proposition 3.1), and so a ∈ Z ( Q, ◦ ). (cid:3) Lemma 3.3.
Let ( Q, · ) be a uniquely -divisible commutative automorphic loop with associ-ated Bruck loop ( Q, ◦ ) . Then a ∈ Z ( Q, ◦ ) if and only if, for all x ∈ Q , P a P x = P x P a . (3.3) Proof.
By Lemmas 2.2 and 3.2, a ∈ Z ( Q, ◦ ) if and only if the identity a ◦ ( x ◦ y ) = x ◦ ( a ◦ y )holds for all x, y ∈ Q . This can be written as [( y ) P x P a ] / = [( y ) P a P x ] / . Squaring bothsides and using unique 2-divisibility to replace y with y , we have ( y ) P x P a = ( y ) P a P x for all x, y ∈ Q . (cid:3) Proofs of the Main Results
Throughout this section, let ( Q, · ) be a uniquely 2-divisible, commutative automorphicloop with associated Bruck loop ( Q, ◦ ). Lemma 4.1. If a ∈ Z ( Q, ◦ ) , then for all x ∈ Q , xL a \ x,a = xL a \ x − ,a . (4.1) Proof.
First, x − = x − L − a − L a − = a − D x − L a − (2.6) = aD x J L a − (2.5) = aD x J D x J L a − (2.6) = aD x D x − L a − = ( x − ) L − a \ x L a − . Thus we compute( x − ) L a \ x,a = ( x − ) L − a \ x L a − L a \ x,a (2.2) = ( x − ) L − a \ x L a \ x,a L a − = ( x − ) L a L − x L a − = aL x − L − x L a − (4.2)= aP x − L a − . ince a − ∈ Z ( Q, ◦ ), we may also apply (4.2) with a − in place of a , and will do so in thenext calculation. Now aP x − L a − = aP x − P a − L a (3.3) = aP a − P x − L a = a − P x − L a (4.2) = ( x − ) L a − \ x,a − (2.6) = ( x − ) L ( a \ x − ) − ,a − (2.4) = ( x − ) L a \ x − ,a , where we used a − ∈ Z ( Q, ◦ ) in the second equality.Putting this together with (4.2), we have ( x − ) L a \ x,a = ( x − ) L a \ x − ,a for all x ∈ Q . Sinceinner mappings are automorphisms, this implies ( xL a \ x,a ) − = ( xL a \ x − ,a ) − . Taking inversesand square roots, we have the desired result. (cid:3) Lemma 4.2. If a ∈ Z ( Q, ◦ ) , then for all x ∈ Q , ( a \ x ) L a \ x − ,a = ( x \ a ) − , (4.3) x − · xP a = a . (4.4) Proof.
We compute( a \ x ) L a \ x − ,a = a \ ( xL a \ x − ,a ) (4.1) = a \ ( xL a \ x,a ) (2.1) = ( a \ x ) L a \ x,a (2.3) = ( x \ a ) − , where we used L a \ x − ,a ∈ Aut Q in the first equality and L a \ x,a ∈ Aut Q in the third equality.To show (4.4), we compute x − · xP a = ( x − ) L a − \ ( ax ) = ( x − ) L a − \ ( ax ) L a − L − ax L ax L − a − = ( a \ ( ax )) − L a − \ ( ax ) ,a − L ax L − a − (2.6) = ( a − \ ( ax ) − ) L a − \ ( ax ) ,a − L ax L − a − (4.3) = (( ax ) − \ a − ) − L ax L − a − (2.6) = (( ax ) \ a ) L ax L − a − = aL − a − = a . Note that in the fifth equality, we are applying (4.3) with a − in place of a and ( ax ) − inplace of x . (cid:3) Lemma 4.3. If a ∈ Z ( Q, ◦ ) , then L a = L ◦ a , and for all integers nL na = L a n . (4.5) Proof.
For x ∈ Q , we compute( a ◦ x ) = ( x ◦ a ) = ( a ) P x (4.4) = xP a L x − P x = x · xP a (2.8) = ( ax ) . Taking square roots, we have a ◦ x = ax , as desired. Then L na = ( L ◦ a ) n (3.2) = L ◦ a n = L a n . (cid:3) Lemma 4.4. If a ∈ Z ( Q, ◦ ) , then for all x ∈ Q , P xa = P x P a . (4.6) Proof.
For each y ∈ Q , yP xa = yP ax = [ ax ◦ y / ] = [( a ◦ x ) ◦ y / ] = [ a ◦ ( x ◦ y / )] = yP x P a , using Lemma 4.3 in the third equality and a ∈ Z ( Q, ◦ ) in the fourth. (cid:3) Lemma 4.5. If a ∈ Z ( Q, ◦ ) , then a ∈ Z ( Q, · ) . roof. We compute L a L x (4.5) = L a L x = L a L a,x L xa (2.4) = L a L a − ,x − L xa = L a L a − L x − L − x − a − L xa (4.5) = L x − L − x − a − L xa (2.6) = L x − L − xa ) − L xa = L x − P xa (4.6) = L x − P x P a = L x L a L − a − (4.5) = L x L a (4.5) = L x L a . By Lemma 2.2, it follows that a ∈ N ρ ( Q, · ), and N ρ ( Q, · ) = Z ( Q, · ) by Proposition 2.1. (cid:3) Lemma 4.6.
Let ( Q, · ) be a uniquely -divisible commutative automorphic loop with associ-ated Bruck loop ( Q, ◦ ) . Then Z ( Q, ◦ ) ⊆ Z ( Q, · ) .Proof. Assume that a ∈ Z ( Q, ◦ ). Then a ∈ Z ( Q, · ) by Lemma 4.5, and thus ( aL x,y ) = a L x,y = a for every x , y ∈ Q . Taking square roots yields aL x,y = a , that is, a ∈ Z ( Q, · ). (cid:3) Now we prove Theorem 1.2, that is, we show that the upper central series of ( Q, · ) and( Q, ◦ ) coincide. Proof of Theorem 1.2.
Since each Z n ( Q ) is the preimage of Z ( Q/Z n − ( Q )) under the canon-ical projection, it follows by induction that it suffices to show Z ( Q, ◦ ) = Z ( Q, · ). Oneinclusion is Lemma 4.6. For the other, suppose a ∈ Z ( Q, · ). Then P a P x = L a L − a − L x L − x − = L x L − x − L a L − a − = P x P a , and so a ∈ Z ( Q, ◦ ) by Lemma 3.3. (cid:3) Proof of Theorem 1.1.
For an odd prime p , let Q be a commutative automorphic p -loop withassociated Bruck loop ( Q, ◦ ). By [8], ( Q, ◦ ) is centrally nilpotent of class, say, n . By Theorem1.2, Q is also centrally nilpotent of class n . (cid:3) From anisotropic planes to automorphic p -loops with trivial nucleus We proved in [12] that a commutative automorphic loop of order p , 2 p , 4 p , p , 2 p or 4 p is an abelian group. For every prime p there exist nonassociative commutative automorphicloops of order p . These loops have been classified up to isomorphism in [6], where theannounced Theorem 1.1 has been used to guarantee nilpotency for p odd.Without commutativity, we do not even know whether automorphic loops of order p areassociative! Nevertheless we show here that the situation is much more complicated thanin the commutative case already for loops of order p . Namely, we construct a family ofautomorphic loops of order p with trivial nucleus.5.1. Anisotropic planes.
Let F be a field, V a finite-dimensional vector space over F , and q : V → F a quadratic form. A subspace W ≤ V is isotropic if q ( x ) = 0 for some 0 = x ∈ W ,else it is anisotropic .It is well known that if F is a finite field and dim V ≥ V must be isotropic. (See[22, Thm. 3.8] for a proof in odd characteristic.) Moreover, if F = GF ( p ) then there is aunique anisotropic space of dimension 2 over F up to isometry. (See [22, Cor. 3.10] for p dd. If p = 2 and V = h x, y i , we must have q (0) = 0, q ( x ) = q ( y ) = q ( x + y ) = 1 for V tobe anisotropic.) Let us call anisotropic subspaces of dimension two anisotropic planes .Since our construction is based on elements of anisotropic planes rather than on the planesthemselves, we will first have a detailed look at anisotropic planes in M (2 , F ), the vectorspace of 2 × F . The determinantdet : M (2 , F ) → F, det (cid:18) a a a a (cid:19) = a a − a a is a quadratic form on M (2 , F ). If F C ⊕ F D is an anisotropic plane in M (2 , F ) then C − ( F C ⊕ F D ) is also anisotropic, and hence, while looking for anisotropic planes, it sufficesto consider subspaces
F I ⊕ F A , where I is the identity matrix and A ∈ GL (2 , F ). Lemma 5.1.
With A ∈ M (2 , F ) , the subspace F I ⊕ F A is an anisotropic plane if and onlyif the characteristic polynomial det( A − λI ) = λ − tr( A ) λ + det( A ) has no roots in F .Proof. The subspace
F I ⊕ F A is anisotropic if and only if det( λI + µA ) = 0 for every λ , µ such that ( λ, µ ) = (0 , A − λI ) = 0 for every λ . Wehave det( A − λI ) = λ − tr( A ) λ + det( A ). (cid:3) We will now impose additional conditions on anisotropic planes over finite fields andestablish their existence or non-existence. We will take advantage of the following strongresult of Perron [18, Thms. 1 and 3] concerning additive properties of the set of quadraticresidues.A nonzero element a ∈ GF ( p ) is a quadratic residue if a = b for some b ∈ GF ( p ). Anonzero element a ∈ GF ( p ) that is not a quadratic residue is a quadratic nonresidue . Theorem 5.2 (Perron) . Let p be a prime, N p the set of quadratic nonresidues, and R p = { a ∈ GF ( p ); a is a quadratic residue or a = 0 } .(i) If p = 4 k − and a = 0 then | ( R p + a ) ∩ R p | = k = | ( R p + a ) ∩ N p | .(ii) If p = 4 k + 1 and a = 0 then | ( R p + a ) ∩ R p | = k + 1 , | ( R p + a ) ∩ N p | = k . Lemma 5.3.
Let p ≥ be a prime. Then there is a quadratic nonresidue a and quadraticresidues b , c such that b − a is a quadratic residue and c − a is a quadratic nonresidue.Proof. Let p = 4 k ±
1. If k ≥ | ( R p − a ) ∩ R p | , | ( R p − a ) ∩ N p | ≥
3. (We need k ≥ b ∈ R p \{ } such that b − a ∈ R p \{ } .)If p = 7 then a = 3, b = 4, c = 1 do the job. If p = 5 then a = 2, b = 1, c = 4 do the job. (cid:3) Lemma 5.4.
Let p be a prime and F = GF ( p ) .(i) There is A ∈ GL (2 , p ) such that tr( A ) = 0 and F I ⊕ F A is anisotropic if and only if p = 2 .(ii) There is A ∈ GL (2 , p ) such that tr( A ) = 0 , det( A ) is a quadratic residue modulo p and F I ⊕ F A is anisotropic if and only if p = 3 .(iii) There is A ∈ GL (2 , p ) such that tr( A ) = 0 , det( A ) is a quadratic nonresidue modulo p and F I ⊕ F A is anisotropic if and only if p = 2 .Proof. Let p ≥
3. For a quadratic nonresidue a and any b ∈ F , let M a,b = (cid:18) − b a − b (cid:19) . ince M a,b = M a, − bI , we have F I ⊕ F M a,b = F I ⊕ F M a, . Now, tr( M a, ) = 0, det( M a, − λI ) = λ − a has no roots, so F I ⊕ F M a,b is anisotropic by Lemma 5.1. Moreover, if b = 0then tr( M a,b ) = − b = 0 and det( M a,b ) = b − a .If p ≥
5, Lemma 5.3 implies that the parameters a and b = 0 can be chosen so thatdet( M a,b ) is a quadratic residue or nonresidue as we please.Let p = 3. Then det( M , ) is a quadratic nonresidue. If tr( A ) = 0 and det( A ) is aquadratic residue then det( A ) = 1 and det( A − λI ) is equal to either λ + λ + 1 (with root1) or λ − λ + 1 (with root − F I ⊕ F A is isotropic.Let p = 2. Then (cid:18) (cid:19) satisfies the conditions of (ii). The only elements A ∈ GL (2 , p ) with tr( A ) = 0 are (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19) , all with det( A + I ) = 0, so F I ⊕ F A is isotropic. There is no matrix satisfying the conditionsof (iii) because there are no quadratic nonresidues in GF (2). (cid:3) Let p be a prime and F = GF ( p ). Call an element A ∈ GL (2 , p ) of an anisotropic plane F I ⊕ F A of type A ) = 0, of type A ) = 0 and det( A ) is a quadratic residue, andof type 3 if tr( A ) = 0 and det( A ) is a quadratic nonresidue.Note that for a fixed prime p we can find elements A of all possible types (with therestrictions of Lemma 5.4) in a single anisotropic plane. This is because we only used matrices A = M a,b with the same a in the proof of Lemma 5.4, and F I ⊕ F M a, = F I ⊕ F M a,b .5.2.
Automorphic loops of order p with trivial nucleus. Let A ∈ GL (2 , p ) be suchthat F I ⊕ F A is an anisotropic plane. Define a binary operation on F × ( F × F ) by( a, x ) · ( b, y ) = ( a + b, x ( I + bA ) + y ( I − aA )) (5.1)and call the resulting groupoid Q ( A ). Since U a = I + aA is invertible for every a ∈ F , we see that Q ( A ) is a loop (see Remark 5.8), and in fact,straightforward calculation shows that( b, y ) L − a,x ) = ( b − a, ( y − xU b − a ) U − − a ) , ( b, y ) R − a,x ) = ( b − a, ( y − xU a − b ) U − a ) . Lemma 5.5.
Let F = GF ( p ) . Let A ∈ GL (2 , p ) be such that F I ⊕ F A is an anisotropicplane in M (2 , p ) . For each z ∈ F × F and each C ∈ GL (2 , p ) satisfying CA = AC , define ϕ z,C : F × ( F × F ) → F × ( F × F ) by ( a, x ) ϕ z,C = ( a, az + xC ) . Then ϕ z,C is an automorphism of Q ( A ) . roof. We compute( a, x ) ϕ z,C · ( b, y ) ϕ z,C = ( a, az + xC ) · ( b, bz + yC )= ( a + b, ( az + xC ) U b + ( bz + yC ) U − a )= ( a + b, ( a + b ) z + xCU b + yCU − a + abzA − abzA )= ( a + b, ( a + b ) z + ( xU b + yU − a ) C )= [( a, x ) · ( b, y )] ϕ z,C , where we have used CA = AC in the fourth equality. Since ϕ z,C is clearly a bijection, wehave the desired result. (cid:3) Proposition 5.6.
Let F = GF ( p ) . Let A ∈ GL (2 , p ) be such that F I ⊕ F A is an anisotropicplane in M (2 , p ) . Then the loop Q = Q ( A ) defined on F × ( F × F ) by (5.1) is an automorphicloop of order p and exponent p with N µ ( Q ) = { (0 , x ) | x ∈ F × F } ∼ = F × F and N λ ( Q ) = N ρ ( Q ) = 1 . In particular, N ( Q ) = Z ( Q ) = 1 and so Q is not centrally nilpotent. Inaddition, if p = 2 then C ( Q ) = Q , while if p > , then C ( Q ) = 1 .Proof. Easy calculations show that the standard generators of the inner mapping group of Q ( A ) are ( b, y ) T ( a,x ) = ( b, ( x ( U − b − U b ) + yU a ) U − − a ) , ( c, z ) R ( a,x ) , ( b,y ) = ( c, ( zU a U b + y ( U − c − a − U − c U − a )) U − a + b ) , (5.2)( c, z ) L ( a,x ) , ( b,y ) = ( c, ( zU − a U − b + y ( U c + a − U c U a )) U − − a − b ) . Since U − b − U b = − bA and U c + a − U c U a = U − c − a − U − c U − a = − caA , we find that eachof these generators is of the form ϕ u,C for an appropriate u ∈ F × F and C ∈ GL (2 , p )commuting with A . Specifically, we have T ( a,x ) = ϕ u,C where u = − xAU − − a and C = U a U − − a ,R ( a,x ) , ( b,y ) = ϕ u,C where u = − ayA U − a + b and C = U a U b U − a + b ,L ( a,x ) , ( b,y ) = ϕ u,C where u = − ayA U − − a − b and C = U − a U − b U − − a − b . Hence Q ( A ) is automorphic by Lemma 5.5.An easy induction shows that powers in Q ( A ) and in F × ( F × F ) coincide, so Q ( A ) hasexponent p .Suppose that ( a, x ) ∈ N µ ( Q ). Then ( c, z ) R ( a,x ) , ( b,y ) = ( c, z ) for every ( c, z ) , ( b, y ). Thus( zU a U b + y ( U − c − a − U − c U − a )) U − a + b = z for every ( c, z ) , ( b, y ). With z = 0, we have y ( U − c − a − U − c U − a ) = − cayA = 0 for every y , hence caA = 0 for every c , and a = 0 follows. On theother hand, clearly (0 , x ) ∈ N µ ( Q ) for every x . We have thus shown N µ ( Q ) = { (0 , x ) | x ∈ F × F } ∼ = F × F .Suppose that ( c, z ) ∈ N λ ( Q ). By Proposition 2.1, N λ ( Q ) = N ρ ( Q ) ≤ N µ ( Q ), so c = 0.We then must have (0 , z ) R ( a,x ) , ( b,y ) = (0 , z ), or zU a U b U − a + b = z , or abzA = 0 for every a , b .In particular, zA = 0 and z = 0. We have proved N λ ( Q ) = 1.If p = 2, then since U a = U − a , it follows that Q is commutative. Now assume that p > a, x ) ∈ C ( Q ). Then x ( U b − U − b ) = y ( U a − U − a ), that is, 2 bxA = 2 ayA for every( b, y ) ∈ Q . With b = 0 we deduce that 2 ayA = 0 for every y , thus 0 = 2 aA , or a = 0. Then2 bxA = 0, and with b = 1 we deduce 2 xA = 0, or x = 0. We have proved that C ( Q ) = 1. (cid:3) emark . The construction Q ( A ) works for every real anisotropic plane R I ⊕ R A andresults in an automorphic loop on R with trivial center. We believe that this is the firsttime a smooth nonassociative automorphic loop has been constructed. Remark . The groupoid Q ( A ) is an automorphic loop as long as I + aA is invertible forevery a ∈ F , which is a weaker condition than having F I ⊕ F A an anisotropic plane, aswitnessed by A = 0, for instance. But we claim that nothing of interest is obtained in themore general case:Let us assume that A ∈ M (2 , F ) is such that I + aA is invertible for every a ∈ F but F I ⊕ F A is not anisotropic. Then det( A ) = 0 and det( A − λI ) = λ − tr( A ) λ = λ ( λ − tr( A ))has no nonzero solutions. Hence tr( A ) = 0 and A = 0. The loop Q = Q ( A ) is still anautomorphic loop by the argument given in the proof of Proposition 5.6, and we claim thatit is a group. Indeed, we have ( c, z ) ∈ N λ ( Q ) = N ( Q ) if and only if ( c, z ) = ( c, z ) R ( a,x ) , ( b,y ) for every ( a, x ), ( b, y ), that is, by (5.2), z = ( zU a U b + y ( U − c − a − U − c U − a )) U − a + b (5.3)for every ( a, x ), ( b, y ). As U b + a − U b U a = − baA = 0 for every a , b , we see that (5.3) holds.6. Open problems
Problem 6.1.
Are the following two statements equivalent for a finite automorphic loop Q ?(i) Q has order a power of .(ii) Every element of Q has order a power of . Problem 6.2.
Let p be a prime. Are all automorphic loops of order p associative? Problem 6.3.
Let p be a prime. Is there an automorphic loop of order a power of p andwith trivial middle nucleus? Problem 6.4.
Let p be a prime. Are there automorphic loops of order p that are notcentrally nilpotent and that are not constructed by Proposition 5.6? Conjecture 6.5.
Let p be a prime and F = GF ( p ) . Let A , B ∈ GL (2 , p ) be such that F I ⊕ F A and
F I ⊕ F B are anisotropic planes. Then the loops Q ( A ) , Q ( B ) constructed by (5.1) are isomorphic if and only if A , B are of the same type. We have verified Conjecture 6.5 computationally for p ≤
5. Taking advantage of Lemma5.4, we can therefore conclude:If p = 2, there is one isomorphism type of loops Q ( A ) obtained from the matrix (cid:18) (cid:19) of type 2—this is the unique commutative automorphic loop of order 8 that is not centrallynilpotent, constructed already in [12]. If p = 3, there are two isomorphism types of loops Q ( A ), corresponding to matrices (cid:18) (cid:19) , (cid:18) (cid:19) of types 1 and 3, respectively. If p = 5, there are three isomorphism types. If Conjecture6.5 is valid for a prime p >
5, then there are three isomorphism types of loops Q ( A ) for thatprime p , according to Lemma 5.4. cknowledgement After this paper was submitted for publication, P. Cs¨org˝o obtained a stronger result thanTheorem 1.1 by her signature technique of connected group transversals. Namely, she proved:
Theorem 6.6 (Cs¨org˝o [5]) . If Q is a finite commutative automorphic p -loop ( p an oddprime), then the multiplication group Mlt Q is a p -group. By a result of Albert [1], Z (Mlt Q ) ∼ = Z ( Q ). In particular, if Mlt Q is a p -group then Z ( Q )is nontrivial. Our Theorem 1.1 then follows from Theorem 6.6 by an easy induction on theorder of Q (as observed by Cs¨org˝o in [5, Cor. 3.2]).Actually, in hindsight it is not difficult to obtain Cs¨org˝o’s Theorem 6.6 from our Theorem1.1: In [23] (see also [21]), Shchukin proved that a commutative automorphic loop Q isnilpotent of class at most n if and only if Mlt Q is nilpotent of class at most 2 n −
1. Nowsuppose Q is a commutative automorphic p -loop, p odd. By Theorem 1.1, Q is nilpotent. Bythe result of Shchukin, Mlt Q is nilpotent, hence a direct product of groups of prime powerorder. Since Z (Mlt Q ) ∼ = Z ( Q ), it follows that Z (Mlt Q ) is a p -group. But then so is Mlt Q .Finally, we are pleased to acknowledge the assistance of Prover9 [17], an automateddeduction tool,
Mace4 [17], a finite model builder, and the
GAP [7] package
Loops [19].
Prover9 was indispensable in the proofs of the lemmas leading up to Theorem 1.2. Weused
Mace4 to find the first automorphic loop of exponent 3 with trivial center in §
5. Weused the
Loops package to verify Conjecture 6.5 for p ≤ References [1] A. A. Albert, Quasigroups, I,
Trans. Amer. Math. Soc. (1943), 507–519.[2] V. D. Belousov, Foundations of the Theory of Quasigroups and Loops , Izdat. Nauka, Moscow, 1967(Russian).[3] R. H. Bruck,
A Survey of Binary Systems , Springer-Verlag, 1971.[4] R. H. Bruck and L. J. Paige, Loops whose inner mappings are automorphisms,
Ann. of Math. (2) (1956), 308–323.[5] P. Cs¨org˝o, Multiplication groups of commutative automorphic p -loops of odd order are p -groups, J.Algebra , to appear.[6] D. A. S. de Barros, A. Grishkov and P. Vojtˇechovsk´y, Commutative automorphic loops of order p ,submitted.[7] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.10 ; 2007,( )[8] G. Glauberman, On loops of odd order I,
J. Algebra (1964), 374–396.[9] G. Glauberman, On loops of odd order II, J. Algebra (1968), 393–414.[10] G. Glauberman and C. R. B. Wright, Nilpotence of finite Moufang 2-loops. J. Algebra (1968), 415–417.[11] P. Jedliˇcka, M. K. Kinyon and P. Vojtˇechovsk´y, The structure of commutative automorphic loops, Trans. Amer. Math. Soc. (2011), 365–384.[12] P. Jedliˇcka, M. K. Kinyon and P. Vojtˇechovsk´y, Constructions of commutative automorphic loops,
Comm. Algebra (2010), no. , 3243–3267.[13] K. W. Johnson, M. K. Kinyon, G. P. Nagy and P. Vojtˇechovsk´y, Searching for small simple automorphicloops, to appear in the London Math. Soc. Journal of Computation and Mathematics.[14] H. Kiechle, The Theory of K-loops , Lecture Notes in Math. , Springer-Verlag, Berlin, 2002.[15] K. Kunen, M. K. Kinyon, J. D. Phillips and P. Vojtˇechovsk´y, The structure of automorphic loops, inpreparation.[16] M. Kinyon, J. D. Phillips and Vojtˇechovsk´y, When is the commutant of a Bol loop a subloop?
Trans.Amer. Math. Soc. (2008), no. 5, 2393–2408.[17] W. McCune,
Prover9 and Mace4 , version 2009-11A, ( )
18] O. Perron,
Bemerkungen ¨uber die Verteilung der quadratischen Reste , Mathematische Zeitschrift (1952), no. , 122–130.[19] G. Nagy and P. Vojtˇechovsk´y, LOOPS: Computing with quasigroups and loops in GAP – a GAP package,version 2.0.0, 2008, ( )[20] H. O. Pflugfelder,
Quasigroups and Loops: Introduction , Sigma Series in Pure Math. , HeldermannVerlag, Berlin, 1990.[21] L. V. Safanova and K. K. Shchukin, On centrally nilpotent loops, Comment. Math. Univ. Carolin. (2000), 401–404.[22] W. Scharlau, Quadratic and Hermitian Forms , A Series of Comprehensive Studies in Mathematics ,Springer-Verlag, Berlin, 1985.[23] K. K. Shchukin, On nilpotency of the multiplication group of an A-loop (Russian),
Mat. Issled. (1988), 116–117.(Jedliˇcka)
Department of Mathematics, Faculty of Engineering, Czech University of LifeSciences, Kam´yck´a 129, 165 21 Prague 6-Suchdol, Czech Republic
E-mail address , Jedliˇcka: [email protected] (Kinyon and Vojtˇechovsk´y)
Department of Mathematics, University of Denver, 2360 S Gay-lord St, Denver, Colorado 80208 USA
E-mail address , Kinyon: [email protected]
E-mail address , Vojtˇechovsk´y: [email protected]@math.du.edu