Download Some classes of finite simple Bol loops

Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Some classes of finite simple Bol loops
LOOPS’07, Prague
Gábor Péter Nagy
Bolyai Institute of the University of Szeged (Hungary)
Marie Curie Fellow at the University of Würzburg (Germany)
August 20, 2007
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Overview
1
Basic concepts on Bol loops
2
Classical simple Bol loops
3
Baer correspondence
4
Simple Bol loops from exact factorizations
5
Finite Bol loops of exponent 2
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Overview
1
Basic concepts on Bol loops
2
Classical simple Bol loops
3
Baer correspondence
4
Simple Bol loops from exact factorizations
5
Finite Bol loops of exponent 2
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Bol loops, Bruck loops and Moufang loops
Right Bol identity: ((xy )z)y = x((yz)y ). (Right) Bol loops have
the right inverse property and are power-associative.
Multiplication on the right behaves good.
Moufang identity: ((xy )z)y = x(y (zy )). Moufang loops are very
closed to groups. The multiplications behave good.
Automorphic inverse property (AIP): (xy )−1 = x −1 y −1 . For
Moufang loops, this implies commutativity. For Bol
loops not. Special case: x 2 = 1 for all x. We call Bol
loops with AIP Bruck loops.
1
2-divisibility: x = y 2 has unique solution y = x 2 . In the finite
case, this is equivalent to odd order.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Isotopy of loops, G-loops
1
The Bol and Moufang properties are universal, that is,
isotopes of Bol or Moufang loops are Bol or Moufang.
2
The Bruck property (=Bol+AIP) is not universal.
Theorem (Robinson, 1968)
Universal Bruck loops are Abelian groups.
Definition: G-loops
The loop Q is a G-loop if it is isomorphic to all its loop isotopes.
Examples
The classical Moufang loops are G-loops. The classical Bol loop is
not a G-loop.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The right multiplication groups of loops
Definition: Right multiplications
Let Q be a loop.
The right multiplication map Rx : L → L of Q is defined by
yRx = yx. They are permutations of Q.
The right multiplication group of Q is
RMlt(Q) = hRx | x ∈ Qi.
It is a transitive permutation groups on Q. The right inner
mapping group RInn(Q) is the stabilizer of 1 in RMlt(Q).
The right section of Q is RSec(Q) = {Rx | x ∈ Q}. It is a
system of coset representatives of each conjugate of RInn(Q)
in RMlt(Q).
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Normal subloops, solvability, simplicity
Definitions
Normal subloop The subloop K ≤ Q is normal if xK = Kx,
x(yK ) = (xy )K , x(Ky ) = (xK )y and (Kx)y = K (xy )
for all x, y ∈ Q. Normal subloops are precisely a) the
kernels of loop homomorphisms, b) the imprimitivity
blocks of the full multiplication group.
Simple loop No proper normal subloops.
Associator-commutator subloop The smallest normal subloop Q 0
of Q such that Q/Q 0 is an Abelian group.
Solvable loop The derived series Q, Q 0 , Q 00 , . . . reaches {1} in a
finite number of steps.
Remark. Q is simple iff Mlt(Q) acts primitively on Q. It is very
hard to characterize simplicity with the right multiplication group!!!
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The existence of finite simple Bol loops
Problem 1 (Orin Chein, LOOPS’99 and LOOPS’03)
Find a finite simple Bol loop which is not Moufang or prove that
no such exists.
Problem 2 (folklore)
Find a finite Bol loop of exponent 2 whose order is not a power of
2 or show that no such exists.
Remark. The smallest Bol loop of exponent 2 whose order is not a
power of 2 is clearly simple. The questions are:
What were the known solutions?
What were the “evidences” for the negative solution?
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Overview
1
Basic concepts on Bol loops
2
Classical simple Bol loops
3
Baer correspondence
4
Simple Bol loops from exact factorizations
5
Finite Bol loops of exponent 2
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Simple groups
The finite simple groups are:
The cyclic groups of prime order.
The alternating groups An , n ≥ 5.
Groups of Lie type: These can be constructed using linear
algebra over any field k. Example: PSL(n, q).
Twisted groups of Lie type: These can be constructed over
fields with an appropriate field automorphisms. Example:
PSU(n, q).
26 sporadic simple groups. Example: The Mathieu groups.
Remark. These are the “main” simple groups in the infinite case,
as well.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The classical Moufang loop
Let k be a field and define the product
a x
c u
ac + x · v
au + dx − y × v
=
,
y b
v d
cy + bv + x × u
y · u + bd
where a, b, c, d ∈ k, x, y, u, v ∈ k 3 . This turns k 8 into an
8-dimensional nonassociative algebra, the split octonions O(k).
Denote by O(k)∗ the set of units and put M(k) = O(k)∗ /k ∗ 1.
Theorems on simple Moufang loops
[Moufang, 1935] O∗ (k), M(k) are (algebraic, differentiable,
compact, etc.) Moufang loops.
[Paige, 1956] M(k)0 is a simple Moufang loop.
[Liebeck, 1987] Any finite simple nonassociative Moufang loop is
isomorphic to M(Fq )0 .
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The classical 2-divisible Bruck loop
Let D = {z ∈ C | |z| < 1} and define
x ⊕y =
x +y
.
1 + x ȳ
Then (D, ⊕, 0) is a simple differentiable 2-divisible Bruck loop.
D has many different interpretations:
1
1
Algebra: The operation y 2 xy 2 on positive definite
Hermitian matrices.
Geometry: The vector addition on the hyperbolic plane.
Physics: Einstein’s addition of relativistic velocity vectors.
Many generalizations.
Glauberman’s Z ∗ -theorem
Finite Bruck loops of odd order are solvable.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The existence of finite simple Bol loops
“Evidences” for the negative solution:
Liebeck’s theorem on finite simple Moufang loops,
Glauberman’s theorem on finite Bruck loops.
The theorem of P.T. Nagy and Strambach (2002) on the
structure of connected compact Bol loops.
Theorem of P.T. Nagy and K. Strambach (2002)
There does not exist any connected compact differentiable simple
proper Bol loop.
Remark. P.T. Nagy and K. Strambach gave a very detailed
description of the structure of connected compact Bol loops.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Overview
1
Basic concepts on Bol loops
2
Classical simple Bol loops
3
Baer correspondence
4
Simple Bol loops from exact factorizations
5
Finite Bol loops of exponent 2
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Loop folders and group envelopes
Definition: Loop folders
The triple (G , H, K ) is a loop folder if
G is a group, H ≤ G , K ⊆ G and 1 ∈ K .
K is a system of coset representatives (=transversal) of each
conjugate of H is G .
Let Q be a loop. Then (RMlt(Q), RInn(Q), RSec(Q)) is a loop
folder. Conversely, let (G , H, K ) a loop folder and define the
operation x ∗ y on K by Hxy = H(x ∗ y ). Then (K , ∗) is a loop.
Definition: Enveloping group of a loop
The group Q is the enveloping group of Q if Q is determined by
some loop folder (G , H, K ).
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The Baer correspondence
Definition: Baer correspondence
The correspondence between loops and loop folders is called the
Baer correspondence.
1
The loop folder of a loop is not unique.
2
Let (G , H, K ) be a loop folder of the loop Q. Assume N C G
is contained in H. Then (G /N, H/N, KN/N) is a loop folder
and the corresponding loop is Q.
3
We say that the loop folder is faithful if G = hK i and the
largest G -normal subgroup coreG (H) of H is trivial.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Some properties of loop folders
1
For a loop folder (G , H, K ), we not necessarily have G = hK i
and coreG (H) = 1.
2
The loop folder (RMlt(Q), RInn(Q), RSec(Q)) of the loop Q
is faithful.
3
Conversely, if (G , H, K ) is a faithful loop folder which
determines the loop Q then G ∼
= RMlt(Q), and H, K
correspond to RInn(Q), RSec(Q), respectively.
4
(G , H, K ) is a Bol loop folder if xyx ∈ K for all x, y ∈ K .
5
For Bol loop folders it suffices to require that K is a system of
right coset representatives for H.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Examples of Bol loop folders
Bol loops are especially suitable for loop folder representations.
Examples of anti-diagonal type
1
(G , {1}, G ) determines the associative group G .
2
Let A be a group and put G = A × A,
H = {(a, 1) | a ∈ A} and K = {(a, a−1 ) | a ∈ A}.
Then (G , H, K ) is a Bol loop folder determining A.
3
Let A be a group of odd order. Put G = A × A,
H = {(a, a) | a ∈ A} and K = {(a, a−1 ) | a ∈ A}.
1
1
Then (G , H, K ) is a 2-divisible Bruck loop folder. (y 2 xy 2 )
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
G-loop property for Bol loop folders
For Bol loops, the G-loop property has a nice interpretation for
loop folders.
Lemma: Loop folders and isotopes
Let (G , H, K ) be a loop folder, `, k ∈ K . Then, (G , H ` , k −1 K ) is a
loop folder and the corresponding loops are isotopes. All isotopes
can be represented in this way.
Theorem: Characterization of G-loops
Let (G , H, K ) be a Bol loop folder. Assume that for all k ∈ K
there is a h ∈ H such that k −1 K = hKh−1 . Then the
corresponding loop Q is a G -loop.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Overview
1
Basic concepts on Bol loops
2
Classical simple Bol loops
3
Baer correspondence
4
Simple Bol loops from exact factorizations
5
Finite Bol loops of exponent 2
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Exact factorizations of groups
Definition: Exact factorization of groups
Let G be a group and A, B ≤ G . We say that AB is an exact
factorization of G if G = AB and A ∩ B = 1.
Finite examples
1
Let G act transitively on X and let H be a regular subgroup.
Then G = Gx H is an exact factorization.
2
G = An , n ≥ 5 odd, H = h(1, 2, . . . , n)i.
3
G acting transitively on X with |X | = p prime and H a
p-Sylow subgroup of G .
Remark. There are many deep results on factorization of simple
groups, see Liebeck, Praeger, Saxl and their references.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The basic construction
Main Theorem 1: Bol loops by exact factorizations
Let A be a group, A = BC an exact factorization and assume that
B, C don’t contain any proper normal subgroup of A. Put
G = A × A, H = B × C ≤ G and K = {(a, a−1 ) | g ∈ A} ⊂ G .
Then (G , H, K ) is a Bol loop folder. The corresponding Bol loop
Q is a G-loop.
Proof. Let a1 , a2 ∈ A be arbitrary. Write a1−1 = b1 c1 , a2 = b2 c2
with bi ∈ B and ci ∈ C . Then the unique element of (a1 , a2 )H ∩ K
is (c1−1 b2−1 , b2 c1 ). This shows that K is a right transversal for H.
Since K is Bol closed, this is enough. With a = bc ∈ A, the
G-property follows from (a, a−1 )K = (b, c −1 )K (b, c −1 )−1 .
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Simplicity
In general, it is rather complicated to characterize simplicity of
loops by loop folders. A special case:
Main Theorem 2: Simplicity of Bol loops
Let A be a noncommutative simple group with nontrivial exact
factorization A = BC . Define the Bol loop folder (G , H, K ) as
above. Then (G , H, K ) is a faithful loop folder and the
corresponding loop Q is a simple proper Bol G-loop.
Proof. Let Q → Q̄ be a surjective homomorphism and (Ḡ , H̄, K̄ ) a
faithful loop folder of Q̄. Then there is a group homomorphism
G → Ḡ mapping H to H̄ and K to K̄ . As the only proper normal
subgroups of G = A × A are A × {1} and {1} × A, the result
follows.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Differentiable and algebraic examples
Differentiable and algebraic examples
1
G = PSL2 (R) acting on the projective line (=circle) and
cos t sin t
H= ±
|t∈R .
− sin t cos t
The loop is a 3-dimensional simple differentiable Bol G -loop.
2
G = SL(2, k) n k 2 acting on the affine plane k 2 and



 1 x1 x2

H =  0 1 x1  | x1 , x2 ∈ k ∼
= k 2.


0 0 1
The loop is a 5-dimensional simple algebraic Bol loop.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Simple Bol loops with solvable group envelopes
Simple Bol loops with solvable group envelopes
Let p, r be primes such that r > p and r | p p − 1. Let R be a
subgroup of F∗pp of order r . Define the transitive permutation
group
G = {z 7→ az τ + b | a ∈ R, b ∈ Fpp , τ ∈ Aut(Fpp )}
acting on Fq . Then |G | = p p+1 r , and G 00 is a regular normal
subgroup of G . Moreover, G has a regular subgroup H 6= G 00 such
that G = G0 H is an exact factorization. The corresponding loop is
a simple Bol loop of order p p+1 r with solvable enveloping group.
Remark. With p = 2 and r = 3 we obtain the smallest known
simple Bol loop of order 24.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Overview
1
Basic concepts on Bol loops
2
Classical simple Bol loops
3
Baer correspondence
4
Simple Bol loops from exact factorizations
5
Finite Bol loops of exponent 2
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Finite Bol loops of exponent 2
Q denotes a nonassociative finite Bol loop of exponent 2 with
G = RMlt(Q). Recall that Q is solvable if and only if its order is a
power of 2.
1
The first construction is due to R.P. Burn (1978). Q has
order 8, and |G | = 16. Q is nilpotent of class 2.
2
[Cameron, Korchmáros, 1993] Aut(Q) cannot act transitively
on Q \ {1}.
3
[Kreuzer, Kolb, 1995] An infinite class of 2-generated loops
with unbounded nilpotency class.
4
[GN, 1998] Q is solvable if and only if G is a 2-group.
5
[Heiss, 1998] Q is solvable if and only if G is solvable.
6
[Kiechle, GN, 2002] A class of nonnilpotent Q’s.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The main breakthrough: Aschbacher’s theorem
Aschbacher’s theorem (2006)
Let Q be a finite nonsolvable Bol loop of exponent 2 with all
proper subloops solvable. Let (G , H, K ) be a faithful loop folder of
Q, J = O2 (G ), and G ∗ = G /J. Then
1 G∗ ∼
= PGL(2, q), with q = 2n + 1 ≥ 5, H ∗ is a Borel subgroup
of G ∗ , and K ∗ consists of 1 and the involutions of G ∗ \ (G ∗ )0 .
2
3
All involutions of G 0 J are contained in J.
Let n0 = |K ∩ J| and n1 = |K ∩ aJ| for a ∈ K \ J. Then n0 is
a power of 2, n0 = n1 2n−1 , and |K | = (q + 1)n0 .
Remark. The smallest case is n = 2, q = 5, G ∗ = PGL(2, 5) ∼
= S5 ,
H ∗ = AGL(1, 5) ∼
= C5 o C4 . (2) implies that G is not a semidirect
product of J and S5 .
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The idea for an exhaustive search
Let Q be the smallest nonsolvable Bol loop of exponent 2. Let
(G , H, K ) be a faithful loop folder of Q, J = O2 (G ), G ∗ = G /J.
1
Q is simple and all subloops are solvable. Aschbacher’s
theorem holds.
2
It is in general true that for N C G , the orbit 1N is a subloop
of Q. Moreover, the orbit of G 0 is Q 0 , hence normal in Q.
3
A5 has no transitive action on 2k points. This implies that
any nonsolvable normal subgroup is transitive on Q.
4
In particular, G has a perfect transitive normal subgroup U:
Take U = G (∞) , then U = U 0 perfect.
5
Scan the Holt-Plesken library of perfect groups satisfying (2).
6
Determine the “good” transitive actions of U on m points and
play with the conjugacy classes of involutions in NSm (U).
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The idea for an exhaustive search
Let Q be the smallest nonsolvable Bol loop of exponent 2. Let
(G , H, K ) be a faithful loop folder of Q, J = O2 (G ), G ∗ = G /J.
1
Q is simple and all subloops are solvable. Aschbacher’s
theorem holds.
2
It is in general true that for N C G , the orbit 1N is a subloop
of Q. Moreover, the orbit of G 0 is Q 0 , hence normal in Q.
3
A5 has no transitive action on 2k points. This implies that
any nonsolvable normal subgroup is transitive on Q.
4
In particular, G has a perfect transitive normal subgroup U:
Take U = G (∞) , then U = U 0 perfect.
5
Scan the Holt-Plesken library of perfect groups satisfying (2).
6
Determine the “good” transitive actions of U on m points and
play with the conjugacy classes of involutions in NSm (U).
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The idea for an exhaustive search
Let Q be the smallest nonsolvable Bol loop of exponent 2. Let
(G , H, K ) be a faithful loop folder of Q, J = O2 (G ), G ∗ = G /J.
1
Q is simple and all subloops are solvable. Aschbacher’s
theorem holds.
2
It is in general true that for N C G , the orbit 1N is a subloop
of Q. Moreover, the orbit of G 0 is Q 0 , hence normal in Q.
3
A5 has no transitive action on 2k points. This implies that
any nonsolvable normal subgroup is transitive on Q.
4
In particular, G has a perfect transitive normal subgroup U:
Take U = G (∞) , then U = U 0 perfect.
5
Scan the Holt-Plesken library of perfect groups satisfying (2).
6
Determine the “good” transitive actions of U on m points and
play with the conjugacy classes of involutions in NSm (U).
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The idea for an exhaustive search
Let Q be the smallest nonsolvable Bol loop of exponent 2. Let
(G , H, K ) be a faithful loop folder of Q, J = O2 (G ), G ∗ = G /J.
1
Q is simple and all subloops are solvable. Aschbacher’s
theorem holds.
2
It is in general true that for N C G , the orbit 1N is a subloop
of Q. Moreover, the orbit of G 0 is Q 0 , hence normal in Q.
3
A5 has no transitive action on 2k points. This implies that
any nonsolvable normal subgroup is transitive on Q.
4
In particular, G has a perfect transitive normal subgroup U:
Take U = G (∞) , then U = U 0 perfect.
5
Scan the Holt-Plesken library of perfect groups satisfying (2).
6
Determine the “good” transitive actions of U on m points and
play with the conjugacy classes of involutions in NSm (U).
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The idea for an exhaustive search
Let Q be the smallest nonsolvable Bol loop of exponent 2. Let
(G , H, K ) be a faithful loop folder of Q, J = O2 (G ), G ∗ = G /J.
1
Q is simple and all subloops are solvable. Aschbacher’s
theorem holds.
2
It is in general true that for N C G , the orbit 1N is a subloop
of Q. Moreover, the orbit of G 0 is Q 0 , hence normal in Q.
3
A5 has no transitive action on 2k points. This implies that
any nonsolvable normal subgroup is transitive on Q.
4
In particular, G has a perfect transitive normal subgroup U:
Take U = G (∞) , then U = U 0 perfect.
5
Scan the Holt-Plesken library of perfect groups satisfying (2).
6
Determine the “good” transitive actions of U on m points and
play with the conjugacy classes of involutions in NSm (U).
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The idea for an exhaustive search
Let Q be the smallest nonsolvable Bol loop of exponent 2. Let
(G , H, K ) be a faithful loop folder of Q, J = O2 (G ), G ∗ = G /J.
1
Q is simple and all subloops are solvable. Aschbacher’s
theorem holds.
2
It is in general true that for N C G , the orbit 1N is a subloop
of Q. Moreover, the orbit of G 0 is Q 0 , hence normal in Q.
3
A5 has no transitive action on 2k points. This implies that
any nonsolvable normal subgroup is transitive on Q.
4
In particular, G has a perfect transitive normal subgroup U:
Take U = G (∞) , then U = U 0 perfect.
5
Scan the Holt-Plesken library of perfect groups satisfying (2).
6
Determine the “good” transitive actions of U on m points and
play with the conjugacy classes of involutions in NSm (U).
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
The first construction
We found a group G satisfying:
(A1) G has an elementary Abelian normal subgroup J of order 32
such that G /J ∼
= PGL(2, 5). In particular, G has order
32 · 120 = 3840.
(A2) J is the F2 -permutation module modulo its center.
(A3) [G , G ]/[G , J] ∼
= SL(2, 5) and G splits over [G , G ]J.
Main Theorem 3: The first simple Bol loop of exponent 2
Let G be a group satisfying (A1)-(A3). Then G has a unique
subgroup H and a unique invariant set K consisting of 1 and
involutions such that (G , H, K ) is a Bol loop folder. The
corresponding loop is a simple Bol loop of exponent 2 and order 96.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
An infinite class of (simple) Bol loops of exponent 2
Lemma
There is a PGL(2, 5)-module U over F2 such that
dim(U) = 8.
U has a unique proper PGL(2, 5)-submodule W ; dim(W ) = 4.
U has precisely 3 AGL(1, 5)-submodule W , T1 , T2 .
1
2
3
For any k ∈ N, put G = G n U k , H = H n T1k ,
K = c G ∪ (J0 + W k ).
(G , H , K ) is a nonsimple Bol loop folder of exponent 2.
Modify H , K by playing with the subgroups and conjugacy
classes of G in order to obtain simple Bol loop folders of
exponent 2.
Remark. For k = 1, G contains more that 30 nonisomorphic simple
Bol loops.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Open problems
Problem 1
Find more finite and infinite simple Bol loops. Find (finite or
infinite) simple Bol loops which are neither G-loops nor the
isotopes of Bruck loops.
Problem 2
Classify those almost simple groups T for which a finite Bol loop
folder (G , H, K ) of exponent 2 exists with T ∼
= G /O2 (G ).
Remark. On the one hand, it seems to me that there are too many
finite simple Bol loops of exponent 2. On the other hand, the
structure of their group envelopes seems to be quite restricted.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Offtopic open problems
Problem 3
Find nonclassical simple Moufang loops or show that no such
exists.
Problem BONUS
Find finite projective planes of non prime power order.
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Basic concepts on Bol loops Classical simple Bol loops Baer correspondence Simple Bol G-loops Finite Bol loops of exponent 2
Offtopic open problems
Problem 3
Find nonclassical simple Moufang loops or show that no such
exists.
Problem BONUS
Find finite projective planes of non prime power order.
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