Near-Optimal Expanding Generating Sets for Solvable Permutation Groups

0.18 MB·English

by  V. Arvind

Near-Optimal Expanding Generating Sets for Solvable Permutation
Groups
V. Arvind† ParthaMukhopadhyay∗ PrajaktaNimbhorkar∗ Yadu Vasudev†
2
1
0 Abstract
2
LetG=hSibeasolvablepermutationgroupofthesymmetricgroupS givenasinputbythegener-
n n
a atingsetS. Wegiveadeterministicpolynomial-timealgorithmthatcomputesanexpandinggenerating
J setofsizeO(n2)forG. Moreprecisely,thealgorithmcomputesasubsetT ⊂GofsizeO(n2)(1/λ)O(1)

6 such that the undirectedCayleygraphCay(G,T) is a λ-spectralexpander(the O notationsuppresses
1 logO(1)nfeactors). Asabyproductofourproof,wegetanewexplicitconstructionofeε-biasspacesof

sizeO(npoly(logd))(1)O(1) forthegroupsZn. TheearlierknownsizeboundwaesO((d+n/ε2))11/2
] ε d
C givenby[AMN98].
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∗ChennaiMathematicalInstitute,Siruseri,India.Emails:{partham,prajakta}@cmi.ac.in
†TheInstituteofMathematicalSciences,Chennai,India.Emails:{arvind,yadu}@imsc.res.in
1
1 Introduction
Expandergraphsareofgreatinterestandimportanceintheoreticalcomputerscience,especiallyinthestudy
of randomness in computation; the monograph by Hoory, Linial, and Wigderson [HLW06] is an excellent
reference. A central problem is the explicit construction of expander graph families [HLW06, LPS88]. By
explicit it is meant that the family of graphs has efficient deterministic constructions, where the notion of
efficiency depends upon the application at hand, e.g. [Rei08]. Explicit constructions with the best known
and near optimal expansion and degree parameters (the so-called Ramanujan graphs) are Cayley expander
families[LPS88].
AlonandRoichman,in[AR94],showthateveryfinitegrouphasalogarithmicsizeexpandinggenerating
set using theprobabilistic method. Forany finitegroup Gand λ > 0, they show that with high probability
a random multiset S of size O( 1 log|G|) picked uniformly at random from G is a λ-spectral expander.
λ2
Algorithmically, if G is given as input by its multiplication table then there is a randomized Las Vegas
algorithm for computing S: pick the multiset S of O( 1 log|G|) many elements from G uniformly and
λ2
independently atrandom andcheckindeterministic time|G|O(1) thatCay(G,T)isaλ-spectral expander.
Wigderson andXiaogaveaderandomization ofthisalgorithm in[WX08](alsosee[AMN11]foranew
combinatorialproofof[WX08]). Givenλ > 0andafinitegroupGbyamultiplicationtable,theyshowthat
indeterministic time|G|O(1) amultiset S ofsizeO( 1 log|G|)can becomputed such that Cay(G,T)isa
λ2
λ-spectral expander.
This paper
SupposethefinitegroupGisasubgroupofthesymmetricgroupS andGisgivenasinputbyagenerating
n
set S, and not explicitly by a multiplication table. The question we address is whether we can compute an
O(log|G|)sizeexpanding generating setforGindeterministic polynomial time.
Notice that if we can randomly (or nearly randomly) sample from the group G in polynomial time,
then the Alon-Roichman theorem implies that an O( 1 log|G|) size sample will be an (1−λ)-expanding
λ2
generating setwithhighprobability. Moreover, itispossible tosample efficiently andnear-uniformly from
anyblack-box groupgivenbyasetofgenerators [Bab91].
Thisproblemcanbeseenasageneralizationoftheconstructionofsmallbiasspacesinsay,Fn[AGHP92].
2
It is easily proved (see e.g. [HLW06]), using some character theory of finite abelian groups, that ε-bias
spacesareprecisely expanding generating setsforFn (andthisholdsforanyfiniteabelian group). Interest-
2
ingly,thebestknownexplicitconstructionofε-biasspacesisofsizeeitherO(n2/ε2)[AGHP92]orO(n/ε3)
[ABN+92],whereastheAlon-Roichmantheoremguaranteestheexistenceofε-biasspacesofsizeO(n/ε2).
Subsequently, Azar,MotwaniandNaor[AMN98]gaveaconstruction ofε-biasspacesforfiniteabelian
groups of the form Zn using Linnik’s theorem and Weil’s character sum bounds. The size of the ε-bias
d
spacetheygiveisO((d+n2/ε2)C)wheretheconstantC comesfromLinnik’stheoremandthecurrentbest
knownboundforC is11/2.
Let G be a finite group, and let S = hg ,g ,...,g i be a generating set for G. Theundirected Cayley
1 2 k
graph Cay(G,S ∪S−1) is an undirected multigraph with vertex set G and edges of the form {x,xg } for
i
each x ∈ G and g ∈ S. Since S is a generating set for G, Cay(G,S ∪ S−1) is a connected regular
i
multigraph.
Inthispaperweproveamoregeneralresult. GivenanysolvablesubgroupGofS (whereGisgivenby
n
a generating set) and λ > 0, weconstruct an expanding generating set T for Gof size O(n2)(1)O(1) such
λ
thatCay(G,T)isaλ-spectral expander. Wealso notethat, forageneral permutation groupG ≤ S given
n
e
2
byageneratingset,wecancompute(indeterministicpolynomialtime)anO(nc)(1)O(1) sizegeneratingset
λ
T suchthatCay(G,T)isaλ-spectral expander. Herecisalargeabsolute constant.
Nowweexplainthemainingredients ofourexpanding generating setconstruction forsolvablegroups:
1. LetGbeafinitegroupandN beanormalsubgroupofG. GivenexpandinggeneratingsetsS andS
1 2
forN andG/N respectively suchthatthecorresponding Cayleygraphsareλ-spectralexpanders, we
give a simple polynomial-time algorithm to construct an expanding generating set S for G such that
Cay(G,S)isalsoλ-spectral expander. Moreover, |S|isboundedbyaconstant factorof|S |+|S |.
1 2
2. Wecomputethederivedseriesforthegivensolvable groupG ≤ S inpolynomial timeusingastan-
n
dard algorithm [Luk93]. This series is of O(logn) length due to Dixon’s theorem. Let the derived
seriesforGbeG = G ⊲G ⊲···⊲G = {1}. Assumingthatwealreadyhaveanexpanding gen-
0 1 k
erating setfor each quotient group G /G (whichisabelian) ofsize O(n2), weapply theprevious
i i+1
steprepeatedlytoobtainanexpandinggeneratingsetforGofsizeO(n2). Wecandothisbecausethe
derivedseriesisanormalseries. e
e
3. Finally, we consider the abelian quotient groups G /G and give a polynomial time algorithm
i i+1
to construct expanding generating sets of size O(n2) for them. This construction applies a series
decomposition of abelian groups as well as makes use of the Ajtai et al construction of expanding
generating setsforZ [AIK+90]. e
t
Wedescribethesteps1,2and3inSections2,3and4respectively. Asasimpleapplication ofourmain
result,wegiveanewexplicitconstructionofε-biasspacesforthegroupsZnwhichweexplaininSection5.
d
Thesize ofourε-bias spaces areO(npoly(logn,logd))(1)O(1). Tothebestofourknowledge, theknown
ε
construction of ε-bias space for Zn gives a size bound of O((d+n/ε2))11/2 [AMN98]. In particular, we
d
note that our construction improves the Azar-Motwani-Naor construction significantly in the parameters d
andn.
It is interesting to ask if we can obtain expanding generating sets of smaller size in deterministic poly-
nomial time. For an upper bound, by the Alon-Roichman theorem we know that there exist expanding
generating setsofsizeO( 1 log|G|)foranyG ≤ S ,whichisbounded byO(nlogn/λ2) = O(n/λ2). In
λ2 n
general, givenG, analgorithmic question istoaskforaminimum size expanding generating setforGthat
makestheCayleygraphλ-spectralexpander. e
Inthisconnection,itisinterestingtonotethefollowingnegativeresultthatLubotzkyandWeiss[LW93]
haveshownaboutsolvable groups: Let{G }beanyinfinitefamilyoffinitesolvablegroups {G }suchthat
i i
each G has derived series of length bounded by some constant ℓ. Further, suppose that Σ is an arbitrary
i i
generating set for G such that its size |Σ | ≤ k for each i and some constant k. Then the Cayley graphs
i i
Cay(G ,Σ )donotform afamily ofexpanders. Incontrast, theyalso exhibit aninfinitefamily ofsolvable
i i
groupsin[LW93]thatgiverisetoconstant-degree Cayleyexpanders.
2 Combining Generating Sets for Normal subgroup and Quotient Group
LetGbeanyfinitegroupandN beanormalsubgroupofG(i.e.g−1Ng = N forallg ∈ G). Wedenotethis
byG⊲N⊲{1}. LetA⊂ N beanexpandinggeneratingsetforN andCay(N,A)beaλ-spectralexpander.
Similarly, suppose B ⊂ G such that B = {Nx|x ∈ B} is an expanding generating set for the quotient
group G/N and Cay(G/N,B) is also a λ-spectral expander. Let X = {x ,x ,...,x } denote a set of
1 2 k
b
b
3
distinctcosetrepresentatives forthenormalsubgroupN inG. InthissectionweshowthatCay(G,A∪B)
isa 1+λ-spectralexpander.
2
InordertoanalyzethespectralexpansionoftheCayleygraphCay(G,A∪B)itisusefultoviewvectors
in C|G| as elements of the group algebra C[G]. The group algebra C[G] consists of linear combinations
α g forα ∈ C. Addition inC[G]iscomponent-wise, andclearly C[G]isa|G|-dimensional vector
g∈G g g
spaceoverC. Theproductof α g and β hisdefinednaturally as: α β gh.
P g∈G g h∈G h g,h∈G g h
Let S ⊂ G be any symmetric subset and let M denote the normalized adjacency matrix of the undi-
S
P P P
rected Cayley graph Cay(G,S). Now, each element a ∈ G defines the linear mapM : C[G] → C[G] by
a
M ( α g) = α ga. Clearly,M = 1 M andM ( α g) = 1 α ga.
a g g g g S |S| a∈S a S g g |S| a∈S g g
InordertoanalyzethespectralexpansionofCay(G,A∪B)weconsiderthebasis{xn |x ∈ X,n ∈ N}
P P P P P P
ofC[G]. Theelementu = 1 nofC[G]corresponds totheuniform distribution supported onN.
N |N| n∈N
Ithasthefollowingimportantproperties:
P
1. Foralla ∈ N M (u )= u because Na = N foreacha ∈ N.
a N N
2. For any b ∈ G consider the linear map σ : C[G] → C[G] defined by conjugation: σ ( α g) =
b b g g
α b−1gb. SinceN ⊳Gthelinearmapσ isanautomorphism ofN. Itfollowsthatforallb ∈ G
g g b P
σ (u ) = u .
b N N
P
Now,consider thesubspaces U andW ofC[G]definedasfollows:
U = α x u , W = x β n β = 0, ∀x∈ X
x N n,x n,x
( ! ) ( ! )
xX∈X xX∈X nX∈N (cid:12) Xn
(cid:12)
ItiseasytoseethatU andW areindeedsubspaces ofC[G]. Furthe(cid:12)rmore,wenotethateveryvectorinU is
orthogonal to every vector in W, i.e. U ⊥ W. This follows easily from the fact that xu is orthogonal to
N
x β nwhenever β nisorthogonal tou . Notethat β nisindeedorthogonal to
n∈N n,x n∈N n,x N n∈N n,x
u when β = 0. WeclaimthatC[G]isadirectsumofitssubspaces U andW.
NP n∈N n,x P P
PropositioPn 2.1. Thegroupalgebra C[G]hasadirectsumdecomposition C[G] = U +W.
Proof. Since U ⊥ W, itsuffices tocheck that dim(U)+dim(W) = |G|. Theset {xu | x ∈ X} forms
N
anorthogonal basisforU sinceforanyx 6= y ∈ X,xu isorthogonal toyu . Thecardinalityofthisbasis
N N
is|X|. Letz ,...,z bethe|N|−1vectorsorthogonaltotheuniformdistribution u intheeigenbasis
1 |N|−1 N
for the Cayley graph Cay(N,A). It is easy to see that the set {xz | x ∈ X,1 ≤ j ≤ |N|−1} of size
j
|X|(|N|−1)formsabasisforW.
Wewillnowprovethemainresultofthissection.
Lemma 2.2. Let G be any finite group and N be a normal subgroup of G and λ < 1/2 be any constant.
SupposeAisanexpanding generating setforN sothatCay(N,A)isaλ-spectralexpander. Furthermore,
suppose B ⊆ Gsuch that B = {Nx|x ∈ B}isan expanding generator for thequotient group G/N and
Cay(G/N,B) is also a λ-spectral expander. Then A∪B is an expanding generating set for G such that
Cay(G,A∪B)isa (1+λ)(mbax|A|,|B|)-spectral expander. Inparticular, if|A| = |B|thenCay(G,A∪B)is
|A|+|B|
b
a (1+λ)-spectral expander.1
2
1The sizes of A and B is not a serious issue for us. Since we consider multisets as expanding generating sets, notice that
wealwaysensure|A|and|B|arewithinafactorof2ofeachotherbyscalingthesmallermultisetappropriately. Indeed,inour
constructionwecanevenensurewhenweapplythislemmathatthemultisetsAandBareofthesamecardinalitywhichisapower
of2.
4
Proof. Wewillgivetheproofonlyforthecasewhen|A| = |B|(thegeneralcaseisidentical).
Let v ∈ C[G] be any vector such that v ⊥ 1 and M denote the adjacency matrix of the Cayley graph
Cay(G,A∪B). Our goal is to show that kMvk ≤ 1+λkvk. Notice that the adjacency matrix M can be
2
writtenas 1(M +M )whereM = 1 M andM = 1 M .2
2 A B A |A| a∈A a B |B| b∈B b
P P
Claim 2.3. For any two vectors u ∈ U and w ∈ W, we have M u ∈ U, M w ∈ W, M u ∈ U,
A A B
M w ∈W,i.e.U andW areinvariant underthetransformations M andM .
B A B
Proof. Consider vectors of the form u = xu ∈ U and w = x β n, where x ∈ X is arbitrary.
N n∈N n,x
By linearity, it suffices to prove for each a ∈ A and b ∈ B that M u ∈ U, M u ∈ U, M w ∈ W,
a b a
P
and M w ∈ W. Notice that M u = xu a = xu = u since u a = u . Furthermore, we can write
b a N N N N
M w = x β na = x γ n′, where γ = β and n′ = na. Since γ =
a n∈N n,x n′∈N n′,x n′,x n,x n′∈N n′,x
β = 0itfollows that M w ∈ W. Now, consider M u = ub. Forx ∈ X and b ∈ B the element
n∈N n,xP Pa b P
xbcanbeuniquely writtenasx n ,wherex ∈ X andn ∈N.
b x,b b x,b
P
M u =xu b = xb(b−1u b) =x n σ (u ) =x n u =x u ∈ U.
b N N b x,b b N b x,b N b N
Finally,
M w =x( β n)b = xb( β b−1nb)= x n β n = x γ n∈ W.
b n,x n,x b x,b bnb−1,x b n,x
n∈N n∈N n∈N n∈N
X X X X
Here,wenotethatγ = β andn′ = b(n−1n)b−1. Hence γ = 0,whichputsM w inthe
n,x n′,x x,b n∈N n,x b
subspace W asclaimed.
P
Claim2.4. Letu∈ U suchthatu⊥ 1andw ∈ W. Then:
1.kM uk ≤kuk, 2.kM wk ≤ kwk, 3.kM uk ≤ λkuk, 4.kM wk ≤ λkwk.
A B B A
Proof. Since M is the normalized adjacency matrix of the Cayley graph Cay(G,A) and M is the nor-
A B
malized adjacency matrix ofthe Cayley graph Cay(G,B), itfollows that for any vectors uand w wehave
theboundskM uk ≤ kukandkM wk ≤ kwk.
A B
Now we prove the third part. Let u = ( α x)u be any vector in U such that u ⊥ 1. Then
x x N
α = 0. Now consider the vector u = α Nx in the group algebra C[G/N]. Notice that
x∈X x P x∈X x
u ⊥ 1. Let Mb denote the normalized adjacency matrix of Cay(G/N,B). Since it is a λ-spectral ex-
P B P
pander it follows that kMBbuk ≤ λkuk. Wbriting out MBbu we get MBbu = |B1| b∈B x∈XαxNxb =
b1 α Nx , because xb = x n and Nxb = Nx (as Nbis a normal subgroup). Hence
|B| b∈B x∈X x b b x,b b P P
the norm of the vector 1 b b α Nx is boundedbby λkuk. Eqbuivalently, the norm of the vector
P P |B| b∈B x∈X x b
1 α x isbounded byλkuk. Ontheotherhand,wehave
|B| b∈B x∈X x b P P
b
P P
1 1
M u= bα x u b = α xb b−1u b
B x N x N
|B| |B|
! !
b x b x
X X X X
1 1
= α x n u = α x u
x b x,b N x b N
|B| |B|
! !
b x b x
XX XX
2Inthecasewhen|A|6=|B|,theadjacencymatrixM willbe |A||+A||B|MA+ |A||B+||B|MB.
5
Foranyvector( γ x)u ∈ U itiseasytoseethatthenormk( γ x)u k= k γ xkku k.
x∈X x N x∈X x N x∈X x N
Therefore,
P P P
1
kM uk = k α x kku k ≤ λk α xkku k = λkuk.
B x b N x N
|B|
b x x∈X
XX X
Wenowshowthefourthpart. Foreachx ∈ X itisusefultoconsider thefollowingsubspaces ofC[G]
C[xN]= {x θ n| θ ∈ C}.
n n
n∈N
X
Foranydistinct x 6= x′ ∈ X,sincexN ∩x′N = ∅,vectors inC[xN]havesupport disjoint fromvectorsin
C[x′N]. Hence C[xN] ⊥ C[x′N] which implies that the subspaces C[xN],x ∈ X are pairwise mutually
orthogonal. Furthermore, thematrixM mapsC[xN]toC[xN]foreachx ∈ X.
A
Now, consider any vector w = x( β n) in W. Letting w = x β n ∈ C[xN]
x∈X n n,x x n∈N n,x
for each x ∈ X we note that M w ∈ C[xN] for each x ∈ X. Hence, by Pythogoras theorem we
A x
P P (cid:0)P (cid:1)
have kwk2 = kw k2 and kM wk2 = kM w k2. Since M w = xM β n , it
x∈X x A x∈X A x A x A n∈N n,x
followsthatkM w k = kM β n k ≤ λk β nk= λkw k.
PA x A n∈N n,x P n∈N n,x x (cid:0)P (cid:1)
Puttingittogether, itfollowsthatkM wk2 ≤ λ2 kw k2 = λ2kwk2.
(cid:0)P A (cid:1) Px∈X x
Wenowcompletetheproofofthelemma. Conside(cid:0)rPanyvectorv (cid:1)∈ C[G]suchthatv ⊥ 1. Letv = u+w
whereu∈ U andw ∈ W. Leth,idenote theinnerproductinC[G]. Thenwehave
1 1
kMvk2 = k(M +M )vk2 = h(M +M )v,(M +M )vi
A B A B A B
4 4
1 1 1
= hM v,M vi+ hM v,M vi+ hM v,M vi
A A B B A B
4 4 2
Weconsidereachofthethreesummandsintheaboveexpression.
hM v,M vi = hM (u+w),M (u+w)i = hM u,M ui+hM w,M wi+2hM u,M wi.
A A A A A A A A A A
ByClaim2.3andthefactthatU ⊥ W,hM u,M wi = 0. Thusweget
A A
hM v,M vi = hM u,M ui+hM w,M wi ≤ kuk2+λ2kwk2, fromClaim2.4.
A A A A A A
Byanidentical argument, Claim2.3andClaim2.4implyhM v,M vi ≤ λ2kuk2 +kwk2. Finally
B B
hM v,M vi = hM (u+w),M (u+w)i
A B A B
= hM u,M ui+hM w,M wi+hM u,M wi+hM w,M ui
A B A B A B A B
= hM u,M ui+hM w,M wi
A B A B
≤ kM ukkM uk+kM wkkM wk(byCauchy-Schwarzinequality)
A B A B
≤ λkuk2 +λkwk2, whichfollowsfromClaim2.4
Combiningalltheinequalities, weget
1 (1+λ)2
kMvk2 ≤ 1+2λ+λ2 kuk2+kwk2 = kvk2.
4 4
(cid:0) (cid:1)(cid:0) (cid:1)
Hence,itfollowsthatkMvk ≤ 1+λkvk.
2
6
Notice that Cay(G,A∪B) is only a 1+λ-spectral expander. We can compute another expanding gen-
2
eratingsetS forGfromA∪B,usingderandomized squaring [RV05],suchthatCay(G,S)isaλ-spectral
expander. We describe this step in Appendix A. As a consequence, we obtain the following lemma which
we will use repeatedly in the rest of the paper. For ease of exposition, we fix λ = 1/4 in the following
lemma.
Lemma2.5. LetGbeafinitegroupandN beanormalsubgroupofGsuchthatN = hAiandCay(N,A)
is a 1/4-spectral expander. Further, let B ⊆ G and B = {Nx | x ∈ B} such that G/N = hBi and
Cay(G/N,B) is a 1/4-spectral expander. Then in time polynomial3 in |A| + |B|, we can construct an
expandinggeneratingsetS forG,suchthat|S| = O(|Ab|+|B|)andCay(G,S)isa1/4-spectralexpbander.
b
3 Normal Series and Solvable Permutation Groups
Insection 2, it wasshown how toconstruct an expanding generating set for agroup Gfrom the expanding
generating setsofitsnormalsubgroup N andquotient groupG/N. Inthissection, weapplyittotheentire
normalseriesforasolvable groupG. Moreprecisely, letG ≤ S suchthatG = G ⊲G ⊲···⊲G = {1}
n 0 1 r
isanormalseries forG. ThusG isanormalsubgroup ofGforeachiandhence G isanormal subgroup
i i
of G for each j < i. We give a construction of an expanding generating set for G, when the expanding
j
generating setsforthequotient groupsG /G areknown.
i i+1
Lemma3.1. LetG ≤ S with normal series {G }r asabove. Further, for each ilet B beagenerating
n i i=0 i
set for G /G such that Cay(G /G ,B ) is a 1/4-spectral expander. Let s = max {|B |}. Then in
i i+1 i i+1 i i i
deterministic timepolynomial innandswecancomputeagenerating setB forGsuchthatCay(G,B)is
a1/4-spectral expander and|B| = clogrsforsomeconstantc > 0.
Proof. TheproofisaneasyapplicationofLemma2.5. Firstsupposewehavethreeindicesk,ℓ,msuchthat
G ⊲G ⊲G and Cay(G /G ,S) and Cay(G /G ,T) both are 1/4-spectral expanders. Then notice
k ℓ m k ℓ ℓ m
that wehave the groups G /G ⊲G /G ⊲{1} and the group Gk is isomorphic to Gk/Gm via a natural
k m ℓ m Gℓ Gℓ/Gm
isomorphism. HenceCay(Gk/Gm,S)isalso a1/4-spectral expander, where S isthe imageofS under the
Gℓ/Gm
said natural isomorphism. Therefore, we can apply Lemma2.5 by setting G to G /G and N to G /G
k m ℓ m
togetagenerating setU forG /GbsuchthatCay(G /G ,U)is1/4-spectrbal and|U|≤ c(|S|+|T|).
k m k m
To apply this inductively to the entire normal series, assume without loss of generality, its length to be
r = 2t. InductivelyassumethatinthenormalseriesG= G ⊲G ⊲G ⊲G ···⊲G = {1},foreach
0 2i 2·2i 3·2i r
quotient group G /G we have an expanding generating set of size cis that makes G /G
j2i (j+1)2i j2i (j+1)2i
1/4-spectral. Now, consider the three groups G ⊲G ⊲G and setting k = 2j2i, ℓ =
(2j)2i (2j+1)2i (2j+2)2i
(2j+1)2i andm = (2j+2)2i intheaboveargumentwegetexpandinggeneratingsetsforG /G
2j2i (2j+2)2i
ofsizeci+1sthatmakesit1/4-spectral. Thelemmafollowsbyinduction.
3.1 Solvablepermutation groups
Now we apply the above lemma to solvable permutation groups. Let G be any finite solvable group. The
derived series for G is the following chain of subgroups of G: G = G ⊲G ⊲···⊲G = {1} where, for
0 1 k
eachi,G isthecommutatorsubgroupofG . ThatisG isthenormalsubgroup ofG generatedbyall
i+1 i i+1 i
3Though the lemma holds for any finite group G, the caveat is that the group operations in G should be polynomial-time
computable. SincewefocusonpermutationgroupsinthispaperwewillrequireitonlyforquotientgroupsG = H/N whereH
andN aresubgroupsofSn.
7
elementsoftheformxyx−1y−1 forx,y ∈ G . ItturnsoutthatG istheminimalnormalsubgroup ofG
i i+1 i
such that G /G is abelian. Furthermore, the derived series isalso a normal series. That means each G
i i+1 i
isinfactanormalsubgroup ofGitself. ItalsoimpliesthatG isanormalsubgroup ofG foreachj < i.
i j
Our algorithm will crucially exploit a property of the derived series of solvable groups G ≤ S : By a
n
theorem of Dixon [Dix68], the length k of the derived series of a solvable subgroup of S is bounded by
n
5log n. Thus,wegetthefollowingresultasadirectapplication ofLemma3.1:
3
Lemma3.2. Suppose G ≤ S isasolvable groupwithderivedseriesG = G ⊲G ⊲···⊲G = {1}such
n 0 1 k
that for each i we have an expanding generating set B for the abelian quotient group G /G such that
i i i+1
Cay(G /G ,B )isa1/4-spectral expander. Lets= max {|B |}. Thenindeterministictimepolynomial
i i+1 i i i
in n and s wecan compute agenerating set B for G such that Cay(G,B) is a 1/4-spectral expander and
|B|= 2O(logk)s =(logn)O(1)s.
Given a solvable permutation group G ≤ S by a generating set the polynomial-time algorithm for
n
computing an expanding generating set will proceed as follows: in deterministic polynomial time, we
first compute [Luk93] generating sets for each subgroup {G } in the derived series for G. In or-
i 1≤i≤k
der toapply the abovelemmaitsuffices tocompute anexpanding generating setB forG /G such that
i i i+1
Cay(G /G ,B )is1/4-spectral. Wedealwiththisprobleminthenextsection.
i i+1 i
4 Abelian Quotient Groups
In Section 3, we have seen how to construct an expanding generating set for a solvable group G, from
expandinggeneratingsetsforthequotientgroupsG /G inthenormalseriesforG. Wearenowleftwith
i i+1
the problem of computing expanding generating sets for the abelian quotient groups G /G . We state
i i+1
a couple of easy lemmas that will allow us to further simplify the problem. We defer the proofs of these
lemmastoAppendixB.
Lemma4.1. LetH andN besubgroupsofS suchthatN isanormalsubgroupofH andH/N isabelian.
n
Let p < p < ... < p be the set of all primes bounded by n and e = ⌈logn⌉. Then, there is an onto
1 2 k
homomorphism φfromtheproduct groupZn ×Zn ×···×Zn totheabelian quotientgroupH/N.
pe pe pe
1 2 k
SupposeH andH aretwofinitegroupssuchthatφ :H → H isanontohomomorphism. Inthenext
1 2 1 2
lemmaweshowthattheφ-imageofanexpanding generating setforH ,isanexpanding generating setfor
1
H .
2
Lemma4.2. Suppose H andH aretwofinitegroups suchthatφ : H → H isanonto homomorphism.
1 2 1 2
Furthermore, suppose Cay(H ,S) is a λ-spectral expander. Then Cay(H ,φ(S)) is also a λ-spectral
1 2
expander.
Now,supposeH,N ≤ S aregroupsgivenbytheirgenerating sets,whereN ⊳H andH/N isabelian.
n
ByLemmas4.1and4.2,itsufficestodescribeapolynomial(inn)timealgorithmforcomputinganexpand-
ing generating set ofsize O(n2)for the product group Zn ×Zn ×···×Zn such that the second largest
pe pe pe
1 2 k
eigenvalue of the corresponding Cayley graph is bounded by 1/4. In the following section, we solve this
problem. e
8
4.1 Expanding generating setfortheproduct group
In this section, we give a deterministic polynomial (in n) time construction of an O(n2) size expanding
generating set for the product group Zn ×Zn ×...×Zn such that the second largest eigenvalue of the
pe pe pe
1 2 k
corresponding Cayleygraphisbounded by1/4. e
Consider the following normal series for this product group given by the subgroups K = Zn ×
i pe−i
1
Zn ×...×Zn for 0 ≤ i ≤ e. Clearly, K ⊲K ⊲···⊲K = {1}. This is obviously anormal series
pe−i pe−i 0 1 e
2 k
sinceK = Zn ×Zn ×...×Zn isabelian. Furthermore, K /K = Zn ×Zn ×...×Zn .
0 pe1 pe2 pek i i+1 p1 p2 pk
Sincethelengthofthisseriesise = ⌈logn⌉wecanapplyLemma3.1toconstruct anexpanding gener-
ating set ofsize O(n2)for K in polynomial timeassuming that wecan compute an expanding generating
0
setofsizeO(n2)forZn ×Zn ×...×Zn indeterministicpolynomialtime. Thus,itsufficestoefficiently
p1 p2 pk
computeanO(n2e)-sizeexpanding generating setfortheproduct groupZn ×Zn ×...×Zn .
p1 p2 pk
In [AIKe+90], Ajtai et al, using some number theory, gave a deterministic polynomial time expanding
generating seetconstruction forthecyclicgroupZ ,wheretisgiveninbinary.
t
Theorem 4.3 ([AIK+90]). Let t be a positive integer given in binary as an input. Then there is a deter-
ministic polynomial-time (i.e. in poly(logt) time) algorithm that computes an expanding generating set T
forZ ofsizeO(log∗tlogt),wherelog∗tistheleastpositive integer k suchthatatowerofk 2’sboundst.
t
Furthermore, Cay(Z ,T)isλ-spectralforanyconstant λ.
t
Now,considerthegroupZ . Sincep p ...p canberepresentedbyO(nlogn)bitsinbinary,we
p1p2...pk 1 2 k
applytheabovetheorem(withλ = 1/4)tocomputeanexpandinggenerating setofsizeO(n)forZ
p1p2...pk
in poly(n) time. Let m = O(logn) be a positive integer to be fixed in the analysis later. Consider the
product group M = Zm × Zm × ...Zm and for 1 ≤ i ≤ m let M = Zm−i × Zme−i × ... × Zm−i.
0 p1 p2 pk i p1 p2 pk
Clearly, the groups M form a normal series for M : M ⊲ M ⊲ ··· ⊲ M = {1}, and the quotient
i 0 0 1 m
groups are M /M = Z × Z × ... × Z = Z . Now we compute (in poly(n) time) an
i i+1 p1 p2 pk p1p2...pk
expanding generating set for Z ofsize O(n)using Theorem 4.3. Then, weapply Lemma3.1 tothe
p1p2···pk
above normal series and compute an expanding generating set of size O(n) for the product group M in
0
polynomial time. The corresponding Cayley geraph will be a 1/4-spectral expander. Now we are ready to
describe theexpanding generating setconstruction forZn ×Zn ×...×eZn .
p1 p2 pk
4.1.1 Thefinalconstruction
For 1 ≤ i ≤ k let m be the least positive integer such that pmi > cn (where c is a suitably large con-
i i
stant). Thus, pmi i ≤ cn2 for each i. For each i, Fpmi be the finite field of pmi i elements which can be
i
deterministically constructed in polynomial time since it is polynomial sized. Clearly, there is an onto ho-
momorphismψ fromthegroupZmp1 ×Zmp2 ×...×Zmpk totheadditivegroupofFpm11 ×Fpm22 ×...×Fpmkk.
Thus, if S is the expanding generating set of size O(n) constructed above for Zm × Zm × ... ×Zm, it
p1 p2 pk
follows from Lemma4.2 that ψ(S) is an expanding generator multiset of size O(n) for the additive group
Fpm1 ×Fpm2 ×...×Fpmk. DefineT ⊂ Fpm1 ×Fepm2 ×...×Fpmk tobe any (say, the lexicographically
1 2 k 1 2 k
first)setofcnmanyk-tuplessuchthatanytwotuples(x ,x ,...,x )and(x′,xe′,...,x′)inT aredistinct
1 2 k 1 2 k
inallcoordinates. Thus x 6= x′ forallj ∈ [k]. ItisobviousthatwecanconstructT bypickingthefirstcn
j j
suchtuplesinlexicographic order.
NowwewilldefinetheexpandinggeneratingsetR. Letx = (x ,x ,...,x ) ∈ T andy = (y ,y ,...,y ) ∈
1 2 k 1 2 k
ψ(S). Definevi = (yi,hxi,yii,hx2i,yii,...,hxin−1,yii)where xji ∈ Fpmi andhxji,yiiisthe inner product
i
9
modulo p of the elements xj and y seen as p -tuples in Zmi. Hence, v is an n-tuple and v ∈ Zn. Now
i i i i pi i i pi
defineR = {(v ,v ,...,v )|x ∈ T,y ∈ ψ(S)}.Noticethat|R|= O(n2).
1 2 k
Claim4.4. Risanexpanding generating setfortheproductgroupZn ×Zn ×...×Zn .
ep1 p2 pk
Proof. Let(χ ,χ ,...,χ )beanontrivialcharacteroftheproductgroupZn ×Zn ×...×Zn ,i.e. there
1 2 k p1 p2 pk
is at least one j such that χ is nontrivial. Letω be a primitive pth root of unity. Recall that, since χ is a
j i i i
character there isacorresponding vector β ∈ Zn, i.e. χ : Zn → Cand χ (u) = ωhβi,ui foru ∈ Zn and
i pi i pi i i pi
theinner product intheexponent isamodulo p inner product. Thecharacter χ isnontrivial ifandonly if
i i
β isanonzeroelementofZn.
i pi
Thecharacters(χ ,χ ,...,χ )oftheabeliangroupZn ×Zn ×...×Zn arealsotheeigenvectorsfor
1 2 k p1 p2 pk
theadjacency matrixoftheCayleygraph ofthegroup withanygenerating set. Thus,inorder toprove that
RisanexpandinggeneratingsetforZn ×Zn ×...×Zn ,itisenoughtoboundthefollowingexponential
p1 p2 pk
sum estimate for the nontrivial characters (χ ,χ ,...,χ ) since that directly bounds the second largest
1 2 k
eigenvalue inabsolute value.
E [χ (v )χ (v )...χ (v)] = E [ωhβ1,v1i...ωhβk,vki]
x∈T,y∈ψ(S) 1 1 2 2 k x∈T,y∈ψ(S) 1 k
(cid:12) (cid:12)
(cid:12) (cid:12) = (cid:12)E [ωhq1(x1),y1i...ωhqk(x(cid:12)k),yki]
(cid:12) (cid:12) (cid:12) x∈T,y∈ψ(S) 1 k (cid:12)
(cid:12) (cid:12)
≤ (cid:12)E E [ωhq1(x1),y1i...ωhqk(xk),yk(cid:12)i] ,
(cid:12) x∈T y∈ψ(S) 1 k (cid:12)
(cid:12) (cid:12)
where q (x) = n−1β xℓ ∈ F [x] for β = (β ,β (cid:12),...,β ). Since the character is (cid:12)nontrivial,
i ℓ=0 i,ℓ pi i i,1 i,2(cid:12) i,n (cid:12)
suppose β 6= 0, then q is a nonzero polynomial of degree at most n − 1. Hence the probability that
j j
P
q (x ) = 0, when x is picked from T is bounded by n . On the other hand, when q (x ) 6= 0 the tuple
j j cn j j
(q (x ),...,q (x )) defines a nontrivial character of the group Zm × ... × Zm. Since S is an expand-
1 1 k k p1 pk
ing generating set for the abelian group Zm × ... × Zm, the character defined by (q (x ),...,q (x ))
p1 pk 1 1 k k
is also an eigenvector for Zm × ... × Zm, in particular w.r.t. generating set S. Hence, we have that
p1 pk
E [ωhq1(x1),y1i...ωhqk(xk),yki] ≤ ε, where the parameter ε can be fixed to an arbitrary small constant
y∈S 1 k
b(cid:12)y Theorem 4.3. Hence the above(cid:12) estimate is bounded by n +ε = 1 +ε which can be made ≤ 1/4 by
(cid:12) (cid:12) cn c
c(cid:12)hoosing candǫsuitably. (cid:12)
Tosummarize,Claim4.4alongwithLemmas4.1and4.2directlyyieldsthefollowingtheorem.
Theorem4.5. Indeterministicpolynomial (inn)timewecanconstructanexpanding generating setofsize
O(n2)fortheproduct group Zn ×···×Zn (whereforeachi,p isaprimenumber ≤ n)thatmakesita
p1 pk i
1/4-spectral expander. Consequently, if H and N are subgroups of S given by generating sets and H/N
n
iesabelianthenindeterministic polynomialtimewecancomputeanexpanding generating setofsizeO(n2)
forH/N thatmakesita1/4-spectral expander.
e
Finally,westatethemaintheoremwhichfollowsdirectly fromtheabovetheorem andLemma3.2.
Theorem4.6. LetG ≤ S beasolvablepermutationgroupgivenbyageneratingset. Thenindeterministic
n
polynomial time we can compute an expanding generating set S of size O(n2) such that the Cayley graph
Cay(G,S)isa1/4-spectral expander.
e
On a related note, in the case of general permutation groups we have the following theorem about
computing expanding generating sets.
10