Automorphism groups of simple Moufang loops over perfect fields

0.14 MB·English

by  Gábor P. Nagy

Automorphism Groups of Simple Moufang
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0 Loops over Perfect Fields
0
2

n By GA´BOR P. NAGY ∗
a
J SZTE Bolyai Institute

4 Aradi v´ertanu´k tere 1, H-6720 Szeged, Hungary
2 e-mail: [email protected]

R] PETR VOJTEˇCHOVSKY´ †
G Department of Mathematics, Iowa State University
. 400 Carver Hall, Ames, Iowa, 50011, USA
h
e-mail: [email protected]
t
a
m
February 2, 2008
[

1
v
0 Abstract
0 Let F be a perfect field and M(F) the nonassociative simple Mo-
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ufang loop consisting of the units in the (unique) split octonion algebra
1
0 O(F) modulo the center. Then Aut(M(F)) is equal to G2(F)⋊Aut(F).
7 In particular, every automorphism of M(F) is induced by a semilinear
0 automorphism of O(F). The proof combines results and methods from
/ geometrical loop theory,groupsof Lietypeand composition algebras; its
h
gistbeinganidentificationoftheautomorphismgroupofaMoufangloop
t
a withasubgroupoftheautomorphismgroupoftheassociated groupwith
m triality.
:
v
i 1 Introduction
X
r
a As we hope to attract the attention of both group- and loop-theorists, we take
theriskofbeing trivialattimesandintroducemostofthe backgroundmaterial
carefully, although briefly. We refer the reader to [11], [10], [3] and [7] for a
more systematic exposition.
A groupoid Q is a quasigroup if the equation xy = z has a unique solution
in Q whenever two of the three elements x, y, z ∈ Q are known. A loop is a
quasigroup with a neutral element, denoted by 1 in the sequel. Moufang loop
∗Supportedbythe“Ja´nos BolyaiFellowship”oftheHungarianAcademyofSciences,and
bythegrantsOTKAT029849 andFKFP0063/2001.
†Partially supported by the Grant Agency of Charles University, grant no. 269/2001/B-
MAT/MFF,andbyresearchassistantshipatIowaStateUniversity.
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is a loop satisfying one of the (equivalent) Moufang identities, for instance the
identity ((xy)x)z = x(y(xz)). The multiplication group Mlt(L) of a loop L is
the group generated by all left and right translations x 7→ ax, x 7→ xa, where
a∈L.
Let C be a vector space over a field F, and N : C −→ F a nondegenerate
quadratic form. Define multiplication · on C so that (C, +, ·) becomes a not
necessarily associative ring. Then C =(C, N) is a composition algebra if N(u·
v) = N(u)·N(v) holds for every u, v ∈C. Composition algebras exist only in
dimensions 1, 2, 4 and8, and we speak of anoctonion algebra when dimC =8.
A composition algebra is called split when it has nontrivial zero divisors. By
[11,Theorem1.8.1],thereisauniquesplitoctonionalgebraO(F)overanyfield
F.
∗
WriteO(F) forthesetofallelementsofunitnorminO(F),andletM(F)be
∗ ∗
thequotientofO(F) byitscenterZ(O(F) )={±1}. Sinceeverycomposition
∗
algebra satisfies all Moufang identities, both O(F) and M(F) are Moufang
loops. Paige proved [9] that M(F) is nonassociative and simple (as a loop).
Liebeck [8] used the classification of finite simple groups to conclude that there
arenoothernonassociativefinite simpleMoufangloopsbesidesM(F),F finite.
Liebeck’s proof relies heavily on results of Doro [4], that relate Moufang
loops to groups with triality. Before we define these groups, allow us to say
a few words about the (standard) notation. Let G be a group. Working in
G⋊ Aut(G), when g ∈ G and α ∈ Aut(G), we write gα for the image of g
under α, and [g, α] for g−1gα. Appealing to this convention, we say that α
centralizes g if gα = g. Now, the pair (G, S) is said to be a group with triality
if S ≤Aut(G), S =hσ, ρi∼=S , σ is an involution, ρ is of order 3, G=[G, S],
3
Z(GS)={1}, and the triality equation
[g, σ][g, σ]ρ[g, σ]ρ2 =1
holds for every g ∈G.
We now turn to geometrical loop theory. A 3-net is an incidence structure
N = (P, L) with point set P and line set L, where L is a disjoint union of 3
classes L (i=1, 2, 3) such that two distinct lines from the same class have no
i
pointincommon,andanytwolinesfromdistinctclassesintersectinexactlyone
point. AlinefromtheclassL isusuallyreferredtoasani-line. Apermutation
i
on P is a collineation of N if it maps lines to lines. We speak of a direction
preserving collineation if the line classes L are invariant under the induced
i
permutation of lines.
There is a canonical correspondence between loops and 3-nets. Any loop L
determines a 3-net when we let P =L×L, L ={{(c, y)|y ∈L}|c∈L}, L =
1 2
{{(x, c)|x ∈ L}|c ∈ L}, L = {{(x, y)|x, y ∈ L, xy = c}|c ∈ L}. Conversely,
3
givena3-netN =(P, L)andthe origin1∈P,wecanintroducemultiplication
on the 1-line ℓ through 1 that turns ℓ into a loop, called the coordinate loop of
N. Since the details of this construction are not essential for what follows, we
omit them.
LetN bea3-netandℓ ∈L ,forsomei. Wedefineacertainpermutationσ
i i ℓi
onthe pointsetP (cf. Figure 1). For P ∈P, leta anda be the lines through
j k
2
"uQ2 b3 "u P′ =σℓ1(P)
" "
" "
a " "
"2 "
" " b
P"u"" a3 ""u " 2
Q
3
ℓ
1
Figure 1: The Bol reflection with axis ℓ
1
P such that a ∈L , a ∈L , and {i, j, k}={1, 2, 3}. Then there are unique
j j k k
intersectionpointsQ =a ∩ℓ ,Q =a ∩ℓ . Wedefineσ (P)=b ∩b ,where
j j i k k i ℓi j k
b is the unique j-line through Q , and b the unique k-line through Q . The
j k k j
permutation σ is clearly an involution satisfying σ (L )=L , σ (L )= L .
ℓi ℓi j k ℓi k j
Ifithappenstobethecasethatσ isacollineation,wecallittheBol reflection
ℓi
with axis ℓ .
i
It is clear that for any collineation γ of N and any line ℓ we have σ =
γ(ℓ)
γσ γ−1. HencethesetofBolreflectionsofN isinvariantunderconjugationsby
ℓ
elementsofthecollineationgroupColl(N)ofN. A3-netN iscalledaMoufang
3-net if σ is a Bol reflection for every line ℓ. Bol proved that N is a Moufang
ℓ
3-net if and only if all coordinate loops of N are Moufang (cf. [2, p. 120]).
We are now coming to the crucial idea of this paper. For a Moufang 3-net
N with origin 1, denote by ℓ (i = 1, 2, 3) the three lines through 1. As in
i
[7], we write Γ for the subgroup of Coll(N) generated by all Bol reflections of
0
N, and Γ for the direction preserving part of Γ . Also, let S be the subgroup
0
generatedbyσ ,σ andσ . Accordingto[7],Γisanormalsubgroupofindex
ℓ1 ℓ2 ℓ3
6 in Γ , Γ =ΓS, and (Γ, S) is a groupwith triality. (Here, S is understood as
0 0
a subgroup of Aut(Γ) by identifying σ ∈S with the map τ 7→στσ−1.) We will
always fix σ = σ and ρ = σ σ in such a situation, to obtain S = hσ, ρi as
ℓ1 ℓ1 ℓ2
in the definition of a group with triality.
2 The Automorphisms
LetC beacompositionalgebraoverF. Amapα:C −→C isalinear automor-
phism (resp.semilinear automorphism)ofC ifitisabijectiveF-linear(resp.F-
semilinear)mappreservingthe multiplication, i.e.,satisfying α(uv)=α(u)α(v)
for every u, v ∈C. It is well known that the group of linear automorphisms of
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O(F) is isomorphic to the Chevalley group G (F), cf. [5, Section 3], [11, Chap-
2
ter 2]. The groupof semilinear automorphisms of O(F) is therefore isomorphic
to G (F)⋊Aut(F).
2
Sinceeverylinearautomorphismofacompositionalgebraisanisometry[11,
Section 1.7], it induces an automorphisms of the loop M(F). By [12, Theorem
3.3],everyelementofO(F)isasumoftwoelementsofnormone. Consequently,
Aut(O(F))≤Aut(M(F)).
Anautomorphismf ∈Aut(M(F))willbecalled(semi)linear ifitisinduced
by a (semi)linear automorphism of O(F). By considering extensions of auto-
morphisms of M(F), it was proved in [12] that Aut(M(F )) is isomorphic to
2
G (F ),whereF isthetwo-elementfield. Theaimofthispaperistogeneralize
2 2 2
this result (although using different techniques) and provethat every automor-
phismof Aut(M(F)) is semilinear,providedF is perfect. We reachthis aimby
identifying Aut(M(F)) with a certain subgroup of the automorphism group of
the group with triality associated with M(F).
To begin with, we recall the geometrical characterizationof automorphisms
of a loop.
Lemma 2.1 (Theorem 10.2 [1]) Let L be a loop and N its associated 3-net.
Any direction preserving collineation which fixes the origin of N is of the form
(x, y) 7→ (xα, yα) for some α ∈ Aut(L). Conversely, the map α : L −→ L is
an automorphism of L if and only if (x, y)7→(xα, yα) is a direction preserving
collineation of N.
We will denote the map (x, y)7→(xα, yα) by ϕ .
α
By [7, Propositions 3.3 and 3.4], N is embedded in Γ = ΓS as follows.
0
The lines of N correspond to the conjugacy classes of σ in Γ , two lines are
0
parallel if and only if the corresponding involutions are Γ-conjugate, and three
pairwise non-parallel lines have a point in common if and only if they generate
a subgroup isomorphic to S . In particular, the three lines through the origin
3
of N correspond to the three involutions of S.
As the set of Bol reflections of N is invariant under conjugations by colli-
neations, every element ϕ ∈ Coll(N) normalizes the group Γ and induces an
automorphism ϕb of Γ. It is not difficult to see that ϕ fixes the three lines
through the origin of N if and only if ϕbcentralizes (the involutions of) S.
Proposition 2.2 LetLbeaMoufangloopandN itsassociated3-net. LetΓ be
0
the group of collineations generated by the Bol reflections of N, Γ the direction
preserving part of Γ , and S ∼= S the group generated by the Bol reflections
0 3
whose axis contains the origin of N. Then Aut(L)∼=C (S).
Aut(Γ)
Proof: Pick α∈Aut(L), and let ϕc be the automorphismof Γ induced by the
α
collineation ϕ . As ϕ fixes the three lines through the origin, ϕc belongs to
α α α
C (S).
Aut(Γ)
Conversely, an element ψ ∈ C (S) normalizes the conjugacy class of σ
Aut(Γ)
in ΓS and preserves the incidence structure defined by the embedding of N.
This means that ψ = ϕb for some collineation ϕ ∈ Coll(N). Now, ψ centralizes
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S,thereforeϕfixesthethreelinesthroughtheorigin. Thusϕmustbedirection
preserving, and there is α∈Aut(L) such that ϕ=ϕ , by Lemma 2.1.
α
It remains to add the last ingredient—groups of Lie type.
Theorem 2.3 Let F be a perfect field. Then the automorphism group of the
nonassociative simple Moufang loop M(F) constructed over F is isomorphic
to the semidirect product G (F)⋊Aut(F). Every automorphism of M(F) is
2
induced by a semilinear automorphism of the split octonion algebra O(F).
Proof: We fix aperfect field F,and assumethat allsimple Moufang loopsand
Lie groups mentioned below are constructed over F.
The groupwith triality associatedwith M turns outto be its multiplicative
groupMlt(M)∼=D ,andthegraphautomorphismsofD areexactlythetriality
4 4
automorphisms of M (cf. [5], [4]). To be more precise, Freudenthal proved this
forthe realsandDoroforfinite fields, howeverthey basedtheir argumentsonly
ontherootsystemandparabolicsubgroups,andthatiswhytheirresultisvalid
over any field.
By [5], C (σ) = B , and by [8, Lemmas 4.9, 4.10 and 4.3], C (ρ) = G .
D4 3 D4 2
As G <B , by [6, p. 28], we have C (S )=G .
2 3 D4 3 2
Since F is perfect, Aut(D ) is isomorphic to ∆⋊(Aut(F)×S ), by a result
4 3
ofSteinberg(cf.[3,Chapter12]). Here,∆isthegroupoftheinneranddiagonal
automorphismsofD ,andS isthegroupofgraphautomorphismsofD . When
4 3 4
char F = 2 then no diagonal automorphisms exist, and ∆ = Inn(D ). When
4
char F 6= 2 then S acts faithfully on ∆/Inn(D ) ∼= C ×C . Hence, in any
3 4 2 2
case, C (S ) = C (S ). Moreover, for the field and graph automorphisms
∆ 3 D4 3
commute, we have C (S )=C (S )⋊Aut(F).
Aut(D4) 3 D4 3
We have proved Aut(M) ∼= G ⋊Aut(F). The last statement follows from
2
the fact that the groupof linear automorphisms of the split octonion algebrais
isomorphic to G .
2
One of the open questions in loop theory is to decide which groups can be
obtained as multiplication groups of loops. Thinking along these lines we ask:
Which groups can be obtained as automorphism groups of loops? Theorem 2.3
yields a partialanswer. Namely, every Lie group of type G over a perfect field
2
can be obtained in this way.
Finally, the former author asked the latter one at the Loops ’99 conference
whetherAut(M(F))issimplewhenF isfinite. Wenowknowthatthishappens
if and only if F is a finite prime field of odd characteristic.
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