Groups equal to a product of three conjugate subgroups

0.24 MB·English

by  John Cannon

GROUPS EQUAL TO A PRODUCT OF THREE CONJUGATE
SUBGROUPS
5
1
JOHNCANNON,MARTINOGARONZI,DANLEVY,ATTILAMARO´TI,
0
ANDIULIANI.SIMION
2

n
a Abstract. Let G bea finite non-solvable group. We provethat there exists
J a proper subgroup A of G such that G is the product of three conjugates
of A, thus replacing an earlier upper bound of 36 with the smallest possible
2
value. Theproofreliesonanequivalentformulationintermsofdoublecosets,
2
and uses the following theorem which is of independent interest and wider

scope: Any group G with a BN-pair and a finite Weyl group W satisfies
R] G=(Bn0B)2 =BBn0B where n0 isany preimageof the longest element of
W. The proof of the last theorem is formulated in the dioid consisting of all
G
unions of double cosets of B in G. Other results on minimal length product
. covers ofagroupbyconjugates ofapropersubgrouparegiven.
h
t
a
m
[
1. Introduction

1
v This paper continues and expands the study, initiated in [15], ofminimal length
6 factorizations of groups in terms of conjugates of a proper subgroup. Here we do
7 not limit the discussion to finite groups and therefore we define γ (G), for any
cp
6
groupG, to be the minimal integer k such that G is a product of k conjugates of a
5
proper subgroup of G, and γ (G)=∞ if no such k exists. Note that any G such
0 cp
. that every maximal subgroup of G is normal, has γ (G) = ∞, so in particular,
1 cp
if G is finite, γ (G) = ∞ if and only if G is nilpotent. In [15] it was shown that
0 cp
5 γcp(G) cannot be bounded by a universal constant in the class of finite solvable
1 non-nilpotent groups while γ (G) ≤ 36 for any finite non-solvable G. In the
cp
: present paper we prove that γ (G)=3 for any finite non-solvable G (Theorem 2
v cp
i below). Note that for any non-trivial G we have γ (G)≥3.
X cp
Oncetheproblemisreducedtothealmostsimplecase,theproofofγ (G)≤36
cp
r
uses two related ingredients which we would like to recall. The first one is the
a
interpretationoftheconditionG=A ···A ,whereA isaconjugateofA<Gfor
1 k i
any1≤i≤k,intermsoftherightmultiplicationactionofGonΩ :={Ax|x∈G}
A
(which induces a right multiplication action of any subgroup of G on Ω ). The
A
secondoneisarelationbetweenproductsofconjugatesofAandproductsofdouble
cosets of A. Our first result can be viewed as an extension of these observations.
Date:January26,2015.
2000 Mathematics Subject Classification. 20D40,20E42,20D05, 20B40.
Key words and phrases. doublecosets,triplefactorization,BN-pairs,dioid.
A.M. acknowledges the support of an Alexander von Humboldt Fellowship for Experienced
ResearchersandthesupportofOTKAK84233.
I.S. acknowledges the support of the Swiss National Science Foundation (project number
P2ELP2 148913) as well as the support of the University of Padova by grants CPDR131579/13
andCPDA125818/12 .
1
2
We denote x−1Ax by Ax for any A≤G and x∈G, and for any two sets A and B,
”A intersects B” is equivalent to A∩B 6=φ.
Theorem 1. Let G be a group and A≤G.
1. Let x,y ∈ G. Then G = AAxAy implies y ∈ AAx. Moreover, assuming that
y ∈AAx, the following three conditions are equivalent:
(a) G=AAxAy.
(b) G=(AzA)(AwA) where z =x−1 and w=xy−1.
(c) G=AAxA= Ax−1A (AxA).
2. Let x∈G then:(cid:0) (cid:1)
(i) G=AAxA if and only if Ox intersects every orbit of the action of A on Ω ,
A
where O is the unique A-orbit satisfying Ax−1 ∈ O. Furthermore, Ox intersects
every orbit of the action of A on Ω if and only if Oz intersects every orbit of the
A
action of A on Ω , where z ∈AxA is arbitrary.
A
(ii) G=(AxA)2 ⇐⇒G=AAxAx2.
Theorem 2. Let G be a finite non-solvable group. Then there exists A < G and
x∈G such that G=(AxA)2. In particular γ (G)=3.
cp
Remark 1. We do not know if for a general group G the existence of z,w ∈ G
such that G=(AzA)(AwA) implies the existence of x∈G such that G=(AxA)2,
or even if the seemingly weaker implication which states that γ (G) = 3 implies
cp
the existence of A<G and x∈G such that G=(AxA)2, is true.
Corollary 1. Any finite group which is not a cyclic p-group has a factorization of
the form G=ABA, where A and B are proper subgroups of G.
A major part of the proof of Theorem 2 relies on the next theorem, which is,
however, of wider scope and interest. We recall that a group G is said to be a
group with a BN-pair ([11], Section 2.1) if there exist subgroups B and N of G
such that: (i) G = hB,Ni; (ii) H := B ∩N E N; (iii) W := N/H (the Weyl
group of the BN-pair) is generated by a set of elements s , i∈I, where I is some
i
indexing set, and s2 = 1 for all i ∈ I; (iv) For any n ∈ N such that s = n H, it
i i i i
holds that n Bn 6= B; (v) n Bn ⊆ Bn nB∪BnB for all n ∈ N and n ∈ N such
i i i i i
that s = n H. By assumption, any element of W can be written as a product of
i i
someofthe s . Wedefinethelengthfunctionl :W →N ([11],Section2.1),bythe
i 0
conditions l(1 ) = 0 and for any 1 6= w ∈ W, the positive integer l(w) is the
W W
minimal length of an expression for w as a product of the s . If W is finite there
i
exists a (unique) element w ∈ W such that l(w ) > l(w) for all w ∈ W, w 6= w
0 0 0
([11], Proposition 2.2.11).
Theorem3. LetGbeagroupwithaBN-pairandafiniteWeylgroup. Letn ∈N
0
be such that n H =w . Then G=(Bn B)2 =BBn0B.
0 0 0
Remark 2. If W is infinite then G is not equal to any (finite) product of double
cosets of B.
Corollary 2. Let G be a group with a BN-pair and a finite Weyl group W. Let
U ≤B satisfy B =UH. Then G=HUUn0U.
Note that in particular, Corollary 2 applies to any group with a split BN-pair
([11], Section 2.5). For the origin, applications and discussion of necessary and
3
sufficient conditions for the identity G = HUUn0U in the context of Chevalley
groups, we refer the reader to [30].
Inthecaseofsimple groupsofLietype,the proofofTheorem3canbe obtained
from [20] where the structure constants of Iwahori-Hecke algebras are described.
Thestartingpointforthecharacterizationoftheseconstantsis[4],whichdescribes
theproductofdoublecosetsforBorelsubgroupsinthecontextofconnectedreduc-
tivelinearalgebraicgroups. AlthoughinprovingTheorem2wedomakeuseofthe
Heckealgebraapproachwhichrepresentsdoublecosetsbygroupalgebrasums,and
then computes the structure constants, the proof of Theorem 3 takes advantage
of the fact that one only needs to distinguish between zero and non-zero structure
constants. This leads us to consider another algebraic structure, which we call the
DC dioid, to handle double cosets. Both approaches are introduced and discussed
in Section 2.
A special case of Theorem 3 is the case where G is a connected reductive lin-
ear algebraic group (see for example [[25], Theorem 11.16]). In this case one can
offer a different proof which is based on a topological argument. This argument
is applicable to any group G such that G is a topological space, and right and
left multiplications by fixed elements of G are continuous maps (such a group is
customarily called a semi-topological group). The core of the argument relies on
standard topological considerations, and it is clear and easy to grasp, so we hope
it can be applied for other groups as well.
Proposition 1. Let G be a semi-topological group. If A is an open dense subset
of G then G=AA−1. In particular, if A=A−1 then G=A2.
Corollary 3. Let G be a connected reductive linear algebraic group, B a Borel
subgroup of G, and n as in Theorem 3. Then, since Bn B is open and dense in
0 0
G, G=(Bn B)2.
0
Returning to the setting of Theorem 3, let G be a group with BN-pair and
finite Weyl group W. For any subset I′ ⊆ I, the standard parabolic subgroup
W of W is defined by W := hs |i∈I′i. There is a bijection ([10], Section 8.3)
I′ I′ i
betweenovergroupsofB(thestandardparabolicsubgroupsofGw.r.tBandN)and
standardparabolicsubgroupsofW givenbyW ←→BW B (hereW isregarded
I′ I′ I′
as a union of left cosets of H in N). Furthermore, since any standard parabolic
subgroup P of G contains B, it is immediate from Theorem 3 that G = PPn0P.
Since W is a finite Coxeter group (see [11] Chapter 2) it is natural to consider the
question of which irreducible finite Coxeter groups (the terminology is explained
in Section 6) are equal to a product of three conjugate parabolic subgroups. This
is answered by Theorem 4. Note that by the above mentioned bijection between
the standard parabolics of W and those of G, to any factorization of W by three
conjugates of some W there is a naturally corresponding factorization of G (see
I′
Remark 3 below).
Theorem 4. Let C be a finite irreducible Coxeter group. Then C is the product
of three conjugates of at least one of its proper parabolic subgroups if C is of one of
the following types: A , n≥2, B , n≥3, D , n≥4, H , E , E , E . Otherwise,
n n n 4 6 7 8
for the remaining types A , B , F , H , I (m>4), C is not a product of three
1 2 4 3 2
conjugates of a proper parabolic subgroup.
Remark 3. Let G be a group with BN-pair and a finite Weyl group W whose
generators are indexed by I. Suppose that W is the product of three conjugates of
4
the standard parabolic subgroup W where I′ ⊆ I. Then G is the product of the
I′
corresponding three conjugates of the standard parabolic subgroup P =BW B.
I′ I′
Finally, we mention other recentpapers which discuss similar problems to those
addressed in the present paper. Liebeck, Nikolov and Shalev ([21]) consider prod-
ucts of conjugate subgroups in finite simple groups. They conjecture that there
exists a universalconstantc such that for everynon-trivialsubgroupA of a simple
group G, the minimal number of conjugates of A such that the setwise product of
these conjugates is G, is bounded above by clog|G|/log|A|. Later, in [22], they
extend this conjecture to subsets of G of size at least 2. For a discussion of triple
factorizationsoftheformG=ABAforA,B <G,see[1]and[2]andthereferences
therein. Liebeck and Pyber have proved ([24] Theorem D) that every finite simple
group of Lie type is a product of no more than 25 Sylow p-subgroups, where p is
the defining characteristic. In [3] it is claimed, without proof, that the 25 can be
replacedby5,whileasketchofaproofofthisclaimforexceptionalLietypegroups
appears in a survey by Pyber and Szabo´ ([28] Theorem 15). For (non-twisted)
Chevalleygroupsthe bestpossibleboundof4is shownto holdbySmolensky,Sury
and Vavilov in [29].
2. Products of Double Cosets
In this section we prove Theorem 1 and describe two algebraic frameworks for
discussing products of double cosets.
LetGbeanygroupandA≤G. RecallthatthesetofalldoublecosetsofAinG
forms a partition of G and AxA=Ax′A if and only if x′ ∈AxA. By a product of
two double cosets of A we mean their setwise product, and (AzA)(AwA) is equal
to a disjoint union of some double cosets of A.
The following lemma is needed for the proof of Theorem 1 and, in fact, gives a
more general version of part of its claims.
Lemma 1. Let G be a group, A≤G and A ,A ,...,A are k ≥3 conjugates of A
1 2 k
in G. Then G = A A ···A if and only if G = (Ax A)(Ax A)···(Ax A) for
1 2 k 1 2 k−1
some x ,...,x ∈G.
1 k−1
Proof. 1. Suppose that G=(Ax A)(Ax A)···(Ax A) for some x ,...,x ∈
1 2 k−1 1 k−1
G. Then
G=(Ax A)(Ax A)···(Ax A)=Ax Ax A···Ax A=
1 2 k−1 1 2 k−1
=AAx−11A(x1x2)−1···A(x1x2···xk−2)−1A(x1x2···xk−1)−1x1x2···xk−1,
and multiplying both sides on the right by (x x ···x )−1, gives
1 2 k−1
G=AAx−11A(x1x2)−1···A(x1x2···xk−2)−1A(x1x2···xk−1)−1.
2. SupposethatG=A A ···A whereeachA isaconjugateofAinGandk ≥3.
1 2 k i
Let gi ∈G, 1≤i≤k be such that Ai =Agi. Then:
G=A A ···A = g−1Ag g−1Ag g−1A···g g−1Ag .
1 2 k 1 1 2 2 3 k−1 k k
Multiplying by g on the left(cid:0)and g−1(cid:1)o(cid:0)n the rig(cid:1)ht, gives: (cid:0) (cid:1)
1 k
G=Ag g−1Ag g−1A···g g−1A=Ag g−1AAg g−1AA···AAg g−1A=
1 2 2 3 k−1 k 1 2 2 3 k−1 k
= Ag g−1A Ag g−1A ··· Ag g−1A .
1 2 2 3 k−1 k
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
5
Proof of Theorem 1. 1. Firstweshowthat(a)=⇒(b)independentoftheassump-
tion y ∈ AAx: Take k = 3 in Lemma 1, and in part (2) of its proof, A = A,
1
g = 1, g = x and g = y. We get (b) for z = x−1 and w = xy−1. Now let
1 2 3
z,w be any two elements of G such that G = (AzA)(AwA). Then, in particular,
1 ∈ (AzA)(AwA), from which it follows that there exist a ,a ∈ A such that
G 1 2
(a wa )−1 = a−1w−1a−1 ∈ AzA. It follows that w−1 ∈ AzA, and so there exist
1 2 2 1
a ,a ∈ A such that w−1 = a za . Therefore, since (a) =⇒ (b), we get that (a)
3 4 3 4
implies yx−1 = a x−1a which is equivalent to y = a x−1a x ∈ AAx. Moreover,
3 4 3 4
since (b) implies w−1 ∈ AzA and hence AwA = Az−1A, substituting z = x−1 in
(b) gives (c), so (b) implies (c). Finally we prove (c) =⇒ (a) for any y ∈ AAx.
Assume G=AAxA, and let y =a x−1a x, where a ,a ∈A are arbitrary. Then
3 4 3 4
AAxAy = AAxAa3x−1a4x =Ax−1Ax x−1a x −1Ax−1a x=
4 4
= Ax−1AxAx−1a x=AAxA(cid:0) x−1a x(cid:1) =G.
4 4
2. (i) We have O = Ax−1 A and therefore A(cid:0)Ax = O(cid:1)x, and hence, since G
is the union of all right(cid:0)cosets(cid:1)of A, the condition G = (AAx)A is equivalent to
(Ox)A = Ω . The last equality is equivalent to the statement that Ox intersects
A
every orbit of the action of A on Ω , since Ω is the union of all of the orbits
A A
of A on Ω . Finally, if z ∈ AxA then z = a xa , with a ,a ∈ A. So Oz =
A 1 2 1 2
Ax−1 Aa xa = Ax−1 Axa = (Ox)a , and Ox intersects every orbit of the
1 2 2 2
(cid:0)action(cid:1)of A on ΩA i(cid:0)f and o(cid:1)nly if (Ox)a2 does.
(ii) First note (by taking inverses) that G= (AxA)2 ⇐⇒G= Ax−1A 2. Now
take k =3 and x =x =x−1 in the proof of Lemma 1 part 1. Th(cid:0)us G=(cid:1)(AxA)2
1 2
implies G = AAxAx2. Conversely, if G = AAxAx2, we take k = 3, g = 1,
1
g = x and g = x2 in the proof part 2 of Lemma 1 and use the above mentioned
2 3
equivalence to get G=(AxA)2. (cid:3)
Let G be a group and A<G. We discuss two possible algebraic frameworks to
deal with products of double cosets of A. Fix a set J ⊆ G of representatives of
distinct double cosets of A in G.
2.1. The Hecke algebra of double cosets. For the firstalgebraicframework,it
issufficient,forourpurpose,toassumethatGisfinite. LetQbethefieldofrational
numbers and Q[G] the group algebra of G over Q. For any subset S ⊆ G define
S ∈ Q[G] by S := g. The set e := 1 AjA|j ∈J is linearly independent
j |A|
gP∈S n o
in Q[G], and its elements satisfy the product rule e e = a e , where the
x y xyj j
jP∈J
structureconstantsa (alsocalledintersectionnumbers)arenonnegativeintegers.
xyj
The span of the set e := 1 AjA|j ∈J in Q[G] is the Hecke algebra of the
j |A|
n o
doublecosetsofA-see[18]Chapter1,Section11D.Theadvantageinrepresenting
the double cosets as group algebra sums lies in its relation to the action of G on
Ω . This relation gives a highly non-trivial algorithm to compute the a from
A xyj
the permutation character 1G associated with the action of G on Ω , viewed as a
A A
complex character - see [9] Chapters 2 and 3 for some general backgroundand [8],
[26]forthedetailsofthespecificalgorithmwelateruseinSection5. Thisalgorithm
requiresthe additional assumptionthat the character1G is multiplicity free. Since
A
we areassuming that G is finite, the number r:=|J| ofdistinct double cosets ofA
6
is a natural number that is equal to the rank of the permutation representation of
GonΩ (seeExercise3.2.27of[13]). Itiscustomarytoviewthea asdefiningr
A xyj
square nonnegative integer r×r matrices called the collapsed adjacency matrices,
via (P ) = a . Typically, the rank of the given permutation representation is
y xj xyj
much smaller than its degree |G:A|, and this explains why the above mentioned
algorithm is capable of computing the collapsed adjacency matrices even for the
larger sporadic groups and some of their subgroups. Note that for a given G and
A and x,y ∈ J, the condition G = (AxA)(AyA) is equivalent to the condition
(P ) 6= 0 for all j ∈ J, namely, that the row labeled by x in the matrix labeled
y xj
by y consists entirely of non-zero entries.
2.2. The dioid of double cosets. Our second approach to products of double
cosets of A in G (here G can be any group, not necessarily finite) utilizes the
following algebraic structure.
Definition1. Aquintuple(R,+,·,0,1)whereRisasetand+and·aretwobinary
operations over R, called respectively addition and multiplication, is a semiring if
the following axioms are satisfied:
(a) (R,+,0) is a commutative monoid with an identity element 0.
(b) (R,·,1) is a monoid with an identity element 1.
(c) Multiplication is right and left distributive over addition.
(d) Multiplication by 0 annihilates R, that is 0·a=a·0=0 for all a∈R.
A starred semiring is a semiring equipped with an additional unary operation
denoted by ∗ (no axioms are imposed in the general case). A semiring in which
addition is idempotent, namely, a+a = a for all a ∈ R, is called an idempotent
semiring or a dioid ([17]). Note that every dioid is equipped with a ”ready made”
partial order relation defined by a≤b if and only if a+b=b. A completesemiring
is a semiring with an infinitary sum operation, namely, a is defined for any
i
iP∈I
indexing set I with a ∈ R for every i ∈ I. An infinitary sum reduces to an
i
ordinary finite sum when I is finite and behaves in a natural way with respect to
partitions ofI. Moreover, it isrequiredtosatisfy theleft andright distributivelaws:
a· a = (a·a ) and a ·a= (a ·a) for any a∈R.
i i i i
(cid:18) (cid:19) (cid:18) (cid:19)
iP∈I iP∈I iP∈I iP∈I
One can easily check that the set D of all unions of double cosets of A in G
together with the empty set is a starred, idempotent, complete semiring under
the following: addition is the operation of set union, multiplication is the setwise
product,0istheemptyset(wedefinetheproductoftheemptysetwithanysubset
of G to be the empty set), 1 is the trivial double coset A1 A = A and the star
G
operation maps each non-empty S ∈D to its inverse S−1 := s−1|s∈S and ∅ to
∅. We shall call this structure the DC dioid associated with(cid:8)G and A.(cid:9)Note that
the natural partial order relation defined above for a general dioid amounts in the
caseoftheDCdioidtosetinclusion. Alsonotethatthestaroperationisinvolutive,
that it is an isomorphism of the additive substructure and an anti-isomorphism of
themultiplicativesubstructure,andthataa∗ =1+...forall∅6=a∈D. InSection
3, when computing in the DC dioid, we will use ∪, ⊆ and −1 for, respectively, +,
≤ and ∗.
We set d := AjA, for any j ∈ J. Any element of the DC dioid can then be
j
written as an infinitary sum c d , where for each j ∈ J, c is either 0 or 1
j j j
jP∈J
7
(0,1 ∈ D). In particular G = d , and the product of any two double cosets
j
jP∈J
defines the structure constants c ∈{0,1}⊆ D via d d = c d . Hence, in
xyj x y xyj j
jP∈J
this framework, the existence of two double cosets of G whose product equals G is
equivalent to the existence of x,y ∈ J such that d d = G, which is equivalent to
x y
c =1 for all j ∈J.
xyj
Finally observe the easy connection between the two approaches (in case of G
finite): c =1 if and only if a 6=0.
xyj xyj
3. Groups with a BN-pair
Throughoutthis sectionGis a fixedgroupwitha fixedBN-pairandthe associ-
ated Weyl group W. Recall that W is generated by a set of involutions {s |i∈I}.
i
We work in the DC dioid defined by the double cosets of B in G (see Section 2).
By Proposition 2.1.2 of [11] there is a bijective map between the set of all double
cosets of B in G and the set of all elements of W. Thus we can choose the set J
(see Section 2) of distinct B-double cosets representatives as a subset of N, in the
form J = {n ∈N|n H =w,w ∈W}, where H = B ∩N and |J| = |W|. The
w w
specific choices of the n do not matter since the double cosets of B are uniquely
w
labeled by the Weyl group elements. Hence, to simplify the notation, we write d
w
for the double coset dnw := BnwB. Observe that d−w1 = dw−1 for every w ∈ W
since (BnwB)−1 = Bn−w1B = Bnw−1B. The following lemma is a basic tool for
computing products in the DC dioid, given these settings.
Lemma 2. For all w∈W, and i∈I we have:
d if l(ws )=l(w)+1
(3.1) d d = wsi i
w si (cid:26) dwsi ∪dw if l(wsi)=l(w)−1,
d if l(s w)=l(w)+1
(3.2) d d = siw i
si w (cid:26) dsiw∪dw if l(siw)=l(w)−1.
Proof. Note that (3.1) is obtained from (3.2) by taking the inverse of both sides
of (3.2), and vice versa. Hence it suffices to prove (3.2). If l(s w) = l(w)+1 the
i
claim follows from [11], Proposition 2.1.3 (ii).
Now suppose that l(s w) = l(w) −1. First we prove the claim for the case
i
w = s . Denote n by n . We have d2 = Bn Bn B, and so we have to show
i si i si i i
that Bn Bn B = B ∪Bn B. The l.h.s is contained in the r.h.s, by axiom (v) of
i i i
the definition of a group with a BN-pair (see Section 1). To prove the reverse
inclusion, observe that since n2 ∈ B we get B ⊆ Bn Bn B. On the other hand,
i i i
by the definition of a group with a BN-pair, n Bn 6= B so Bn Bn B 6=B. Since
i i i i
Bn Bn B ⊆B∪Bn B,thisimpliesBn B ⊆Bn Bn B,andaltogetherBn Bn B =
i i i i i i i i
B∪Bn B, which is equivalent to d d =d ∪d .
i si si 1 si
Now consider a general w ∈ W of positive length. Using s2 = 1 and l(s w) =
i i
l(w)−1, we get l(w) = l(s (s w)) = l(s w)+1 and by the upper branch of the
i i i
identity, which is already accounted for, dsidsiw = ds2iw = dw. Multiply the last
relation by d on the left. We get:
si
d d =(d )2d =(d ∪d )d =d ∪d d =d ∪d .
si w si siw 1 si siw siw si siw siw w
(cid:3)
8
An expression for w ∈W as a product of l(w) generators s is called a reduced
i
word for w. Since reduced words are, in general, not unique, we introduce the
following formalism in order to make our arguments precise. Let E be the free
I
monoidgeneratedby the letters {ε |i∈I}. Let ε be the identity of E , and define
i 0 I
a length function l : E → N by l(ε ) = 0, l(ε ) = 1 for all i ∈ I, and recursively
I 0 0 i
l(ε ε ···ε ) = l ε ε ···ε + l(ε ) for all m ≥ 2, where i ,...,i ∈
i1 i2 im i1 i2 im−1 im 1 m
I ∪{0} (assume 0 ∈/(cid:0) I). We deno(cid:1)te both length functions on EI and on W by
the same letter l. Let τ : E → W be the monoid homomorphism defined by
I
τ(ε )=1 andτ(ε )=s foralli∈I. Wewillcalla=ε ε ···ε ∈E reduced
0 W i i i1 i2 im I
if l(a)=l(τ(a)). For each w∈W define:
Red(w):= a∈τ−1(w)|a is reduced ,
(cid:8) (cid:9)
namely,Red(w) isthe setofallreducedexpressionsforw. ClearlyRed(w)6=∅for
any w ∈ W. Finally, recall that if W is finite then w denotes the unique element
0
of maximal length in W.
Lemma 3. Let x,y ∈ E be such that x, y and xy are reduced. Then d =
I τ(xy)
d d .
τ(x) τ(y)
Proof. By induction on l(y). For l(y) = 0 we have y = ε so xy = xε = x and
0 0
τ(y)=τ(ε )=1 . In this case d =d and d d =d B =d so
0 W τ(xy) τ(x) τ(x) τ(y) τ(x) τ(x)
the claim holds. If l(y)= 1 then y =ε with i∈ I. We have d d =d d
i τ(x) τ(y) τ(x) si
andsincexy isreduced,l(xy)=l(τ(xy))=l(τ(x)s ). Sincexisreduced,l(xy)=
i
l(xε ) = l(x)+1 = l(τ(x))+1, and so we proved l(τ(x)s ) = l(τ(x))+1. By
i i
Lemma 2 we get d d =d =d .
τ(x) si τ(x)si τ(xy)
Now suppose that l(y)=m>1. Thus y =ε ε ···ε for some i ,...,i ∈I.
i1 i2 im 1 m
Since y is reduced, l(y) = l(τ(y)), and hence y = ε y′, and l(y′) = l(τ(y′)) =
i1
m−1, so y′ ∈ E −{ε } is reduced of length l(y′) = l(y)−1. By Lemma 2 we
I 0
have d = d d . By assumption, xy = xε y′ is also reduced. Therefore
τ(y) si1 τ(y′) i1
l(τ(xε ))=l(x)+1,because,otherwise,l(τ(xε ))<l(x)+1andsol(τ(xy))=
i1 i1
l(τ(xε )τ(y′)) < l(x)+1+l(y)−1 = l(x)+l(y) = l(xy) contradicting the
i1
fact the xy is reduced. Now we can use the induction assumption with xε in
i1
the role of x and y′ in the role of y. We get d = d = d d .
τ(xy) τ(xεi1y′) τ(xεi1) τ(y′)
Using twice the length 1 case (it works for both orderings) gives d d =
τ(xεi1) τ(y′)
d d d =d d = d d . (cid:3)
τ(x) si1 τ(y′) τ(x) τ(εi1y′) τ(x) τ(y)
Lemma 4. Assume that W is finite. Let b∈E . Then d ⊆d d .
I w0 τ(b) w0
Proof. By induction on l(τ(b)). If l(τ(b)) = 0 then d is the multiplicative
τ(b)
identity ofthe DC dioidand the claimis clear. If l(τ(b))>0 choosea reducedex-
pressionε ε ···ε ∈Red(b), and set b′ =ε ε ···ε . By Lemma 3 we have
i1 i2 im i1 i2 im−1
d = d d = d d . Hence d d = d d d . By maximality
τ(b) τ(b′) τ(εim) τ(b′) sim τ(b) w0 τ(b′) sim w0
of l(w ), we have l(s w )<l(w ), hence, by Lemma 2 (3.2), d ⊆d d and
0 im 0 0 w0 sim w0
so d d ⊆d d d =d d . Since l(τ(b′))=m−1<m=l(τ(b)) we
τ(b′) w0 τ(b′) sim w0 τ(b) w0
have by induction d ⊆d d and the claim follows. (cid:3)
w0 τ(b′) w0
Lemma 5. Let a,b∈E . Then d ⊆d d .
I τ(ab) τ(a) τ(b)
Proof. We have d d = Bn B Bn B = Bn Bn B. Since 1 ∈
τ(a) τ(b) τ(a) τ(b) τ(a) τ(b) G
Bwegetnτ(a)nτ(b) ∈nτ(a)Bn(cid:0)τ(b). Henc(cid:1)e(cid:0)nτ(a)nτ(b(cid:1)) ∈dτ(a)dτ(b)andBnτ(a)nτ(b)B ⊆
d d . Now, τ is a homomorphism, so τ(ab) = τ(a)τ(b), and hence d =
τ(a) τ(b) τ(ab)
9
Bn B = Bn B. Now n H n H = n H, and since H ⊆ B
τ(ab) τ(a)τ(b) τ(a) τ(b) τ(a)τ(b)
this implies Bnτ(a)τ(b)B = Bn(cid:0)τ(a)nτ(b)(cid:1)B(cid:0). Thus(cid:1)dτ(ab) = Bnτ(a)nτ(b)B and the
claim follows. (cid:3)
Note that one can also prove Lemma 4, using Lemma 3 and Lemma 2 (3.2),
without explicitly invoking the definition of the d .
w
Proof of Theorem 3. Let w ∈ W be arbitrary. We will show that d ⊆ d2 . If
w w0
w = 1 we get d = B and d2 = BBn0B since w2 = 1. Since B ⊆ BBn0B our
W w w0 0
claim holds for this case and hence we can assume that l(w)>0.
Fix a∈Red(ww ). Since w 6=1,wehaveww 6=w andhencel(ww )<l(w ).
0 0 0 0 0
Hence there exists b ∈ E such that b is reduced, l(b) > 0 and ab ∈ Red(w )
I 0
(see the first paragraph of the proof of Proposition 2.5.5 of [11] and note that
it relies on properties of W that apply for the general case of a group with a
BN-pair and a finite Weyl group W). We have τ(a) = ww and hence τ(ab) =
0
τ(a)τ(b) =ww τ(b). On the other hand ab ∈ Red(w ) implies that τ(ab)= w ,
0 0 0
so, substituting this in the previous relation, we obtain w = ww τ(b) and hence
0 0
τ(b)=w w−1w . Furthermore, d2 =d d =d d . By Lemma 3, d =
0 0 w0 w0 w0 τ(ab) w0 τ(ab)
d d and hence we proved d2 =d d d .
τ(a) τ(b) w0 τ(a) τ(b) w0
By Lemma 4 we have d ⊆ d d . Hence d d ⊆ d d d = d2 .
w0 τ(b) w0 τ(a) w0 τ(a) τ(b) w0 w0
Now we get from Lemma 5 (taking, in that lemma, b ∈ E such that τ(b) = w )
k 0
d ⊆d d . However, a∈Red(ww ), so τ(a)=ww , giving w =τ(a)w ,
τ(a)w0 τ(a) w0 0 0 0
and so d ⊆d d . Combining this with d d ⊆d2 we obtain d ⊆d2 as
w τ(a) w0 τ(a) w0 w0 w w0
desired. (cid:3)
Proof of Remark 2. An arbitrary product of double cosets of B in G takes the
form, in the associated DC dioid, d ···d for some w ,...,w ∈ W. For each
w1 wm 1 m
1 ≤ j ≤ m, choose a reduced expression w = s ···s . Applying Lemma 3,
j j1 jl(wj)
we get d = d ···d . Thus, any product of double cosets of B in G, is a
wj sj1 sjl(wj)
finite length product of some d for some i ∈ I (with possible repetitions). Now,
si
employing Lemma 2, it follows by induction on the number of the d factors in
si
the product, that such a product is a union of a finite number of various d with
w
w ∈ W. Since G is equal to the union of the infinite set {d |w ∈W}, no product
w
of double cosets of B in G is equal to G. (cid:3)
Proof of Corollary 2. The claim follows immediately fromTheorem3 upon substi-
tuting B = UH in G = BBn0B and noting that H commutes with both U and
n . (cid:3)
0
4. The topological approach
This section illustrates the topological approach to products of double cosets.
Proof of Proposition 1. Let g ∈ G. We have to prove that g ∈ AA−1. Consider
gA. This is an open subset of G, and hence, since A is dense in G, it intersects A,
that is gA∩A6=∅. It follows that there exist a,a′ ∈A such that ga′ =a. This is
equivalent to g =a(a′)−1 ∈AA−1. (cid:3)
Proof of Corollary 3. Observe that a linear algebraic group is a semi-topological
group and hence we can apply Proposition 1 with A = Bn B, provided that we
0
show that (Bn B)−1 = Bn B and that Bn B is an open dense subset of G. The
0 0 0
10
first claim is an immediate consequence of w2 = 1 and the second is well known
0
(see [25], proof of Theorem 11.20). (cid:3)
5. γ (G)=3 for a non-solvable finite G
cp
In this section we prove Theorem 2. For any group G we define β (G) as the
cp
smallest natural number k such that there exist A < G and x ∈ G for which G =
(AxA)k (⇐⇒G=AAx···Axk),andβ (G)=∞ifnosuchkexists. Thenβ (G)
cp cp
satisfiestheliftingproperty,namely,foreveryN EGwehaveβ (G)≤β (G/N)
cp cp
(see Proposition 8 of [15] and note that (xN)i = xiN). Assume, henceforth, that
G is a finite non-solvable group. Following the proof of Theorem 3 of [15], using
the same arguments, we reduce to the case where G has a unique minimal normal
subgroupN, and N is non-abelian. Thus we can assume the minimal non-solvable
setting (see the beginning of Section 3 of [15]). Now one checks that the analysis
carried in Section 3 of [15] goes through also for β (G) since the restriction that
cp
the conjugating elements are successive powers of a given x lifts from subgroups
to direct products of subgroups and to normalizers of subgroups. Thus, relying on
an analogue of Lemma 14 of [15] for β (G), we have the following reduction of
cp
the statement β (G) = 2 for every finite non-solvable group G to almost simple
cp
groups.
Claim1. β (G)=2foreveryfinitenon-solvablegroupGifforanyalmostsimple
cp
group X with S =soc(X), there exist A<S and x∈S such that:
(a) S =(AxA)2 and (b) N (A)S =X.
X
Proof. By the discussion preceding the claim we can assume that G satisfies the
minimal non-solvable setting of Section 3 of [15]. Hence it is sufficient to show
that the existence of A<S with the properties specified in the claim, ensures the
existence of U ≤ X satisfying all of the assumptions of (the β (G) analogue of)
cp
Lemma14of[15]. Indeed,wesetU :=N (A). ThenUS =X isjust(b). Inorder
X
to prove that S = ((U ∩S)x(U ∩S))2 we note that U ∩S = N (A) ≥ A. Using
S
this,thefactthatforanys∈S wehave(N (A))s =N (As),andTheorem1(2)(ii)
S S
we get that S = (AxA)2 = AAxAx2 implies S = N (A)(N (A))x(N (A))x2 =
S S S
((N (A))x(N (A)))2 = ((U ∩S)x(U ∩S))2. Finally, N (A) ≥ A > 1 and
S S S
N (A)<S since S is simple and A is proper in S and non-trivial. (cid:3)
S
Thus it remains to use the classification of simple non-abelian groups in order
to find an appropriate choice of A for any almost simple group X. As a matter of
fact,inallcaseswefindA<S,foreachsimplenon-abeliangroupS,suchthatthis
A satisfies (a) and (b) for all almost simple X with S =soc(X).
1. S ∼= Alt(n), the alternating group of degree n, and n ≥ 5. Here we use
Corollary 15 of [15]. Note that in all cases there, A < S is a point stabilizer of a
2-transitive action of S on some set. In this case S =A∪AxA for any x∈S−A.
Furthermore, one can choose x to be an involution so clearly A⊆(AxA)2. On the
otherhand(AxA)2 6=AsinceAx ⊆(AxA)2 andA6=Ax sinceAisselfnormalizing.
Thus S =(AxA)2 so condition (a) of Claim 1 holds, while condition (b) is checked
in the proof of Corollary 15 of [15].
2. S is a simple group of Lie type. Here we use the fact that all finite simple
groups of Lie type are groups with a BN-pair (See [16] Definition 2.2.8, Theorem
2.2.10and Theorem2.3.4. The Tits groupis not counted in this category). Hence,