On the homology of Borel subgroup of SL(2,Fp)

Research Article

Falcuty of Mathematics and Computer Science, Ho Chi Minh University of Science, Vietnam

Correspondence

Vo Quoc Bao, Falcuty of Mathematics and Computer Science, Ho Chi Minh University of Science, Vietnam Email: voquocbao0603@gmail.com History

• Received: 2018-12-04

• Accepted: 2019-03-22

• Published: 2019-08-19

DOI :

Copyright

© VNU-HCM Press.This is an open- access article distributed under the terms of the Creative Commons Attribution 4.0 International license.

On the homology of Borel subgroup of SL(2,Fp)

Bui Anh Tuan, Vo Quoc Bao

^*

ABSTRACT

In the theory of algebraic groups, a Borel subgroup of an algebraic group is a maximal Zariski closed and connected solvable algebraic subgroup. In the case of the special linear groupSL2over finite fieldsFpthe subgroup of invertible upper triangular matrices B is a Borel subgroup. According to Adem¹, these are periodic groups. In this paper we compute the integral homology of the Borel subgroup B of the special linear groupSL(2,Fp)where p is a prime. In order to compute the inte- gral homology of B, we decompose it intoℓ−primary parts. We compute the first summand based on Invariant Theory and compute the rest based on Lyndon-Hochschild-Serre spectral sequence. In conclusion, we found the presentation ofBand its period. Furthermore, we also explicitly compute the integral homology ofB.

Key words:Ring cohomology of p-groups, periodic groups, Invariant Theory, Lyndon-Hochschild- Serre spectral sequence

PRELIMINARIES

For reference, we briefly recite some facts about group cohomology and the transfer homomorphism^1–4, which will be used frequently throughout this paper.

LetGbe a finite group andAbe aG−module, then we define Hⁿ(G,A):=Hⁿ(B_G,A)

whereB_Gis classifying space of the groupG.The groupHⁿ(G,A)is called thecohomology group of Gwith (untwisted) coefficientA.IfH⊂Gis a subgroup, the inclusionBH →BGinduces a map in cohomology

res^G_H:Hⁿ(G,A)→Hⁿ(H,A)

calledrestriction. Because inner automorphisms ofGact trivially on cohomology, we haveIm(res^H_G)is con- tained inHⁿ(H,A)^NG(H)/H. There is also atransfer mapgoing other way,

tr^G_H:Hⁿ(H,A)→Hⁿ(G,A).

They are related by two composition formulae.

• tr^H_G◦res^G_Hequals multiplication by[G:H]onHⁿ(G,A).

• (Double coset formula)

res^G_H◦tr^GK=

∑

i

tr_H∩x

iKx⁻¹_i ◦

∑

i

res^x_H^iKx_∩x⁻¹ⁱ

iK_i◦cxi

whereK⊂Gis also a subgroup, the sum is over double-coset representatives, andCx:xHx⁻¹→His conju- gation.

Some consequences of the two formulae.

• Ifpdoes not divideG,thenHⁿ( G,Fp

)=0for alln>0

• IfHis contains a Sylowp−subgroup ofG,then https://doi.org/10.32508/stdj.v22i3.1225

res^G_H:Hⁿ(G,A)_(p)→Hⁿ(H,A)_(p)

is injective, where the subscipt isp−primary part.

• IfHcontains a Sylowp−subgroup ofGand is normal inG,then

res^G_H:Hⁿ(G,A)_(p)∼=Hⁿ(H,A)^G/H.

• IfGis an elementary abelianp−group andHis a proper subgroup, then

tr^G_H:Hⁿ(H,A)→Hⁿ(G,A) is zero.

LetGbe a finite group andAbe aG−module, then we define Hn(G,A):=Hn(BG,A)

whereB_Gis classifying space of the groupG.The groupHn(G,A)is called thehomology group of Gwith (untwisted) coefficientA.Taken=1andA=Z,there is a canonical isomorphism

H₁(G,Z)∼=G/[G,G] (1)

where[G,G]is commutator subgroup ofG.

WhenH⊂Gis a normal subgroup, there is a Lyndon-Hochschild-Serre spectral sequence Hp

(G/H,H_q(H,A))

⇒Hp+q(G,A).

The following two facts are the best tool to change of ring or to change between cohomology and homology.

• (Universal coefficient theorem for group homology)

Hp(G,A)∼=(

Hp(G,Z),A)

⊕Tor(

H_p−1(G,Z),A) .

• (Dual coefficient theorem for group cohomology)

H^p(G,A)∼=Hom(

Hp(G,Z),A)

⊕Ext(

Hp−1(G,Z),A) .

THE PRESENTATION AND THE PERIODICITY OF BOREL SUBGROUP OF SL (2, F

P

)

LetG=SL( 2,Fp

).LetBbe the subgroup of upper triangular matrices inG,Dthe subgroup of diagonal ma- trices inG,andUthe subgroup of upper triangular matrices with all their diagonal coefficients equal to1.We describe these group as follow

B={ ( ∗ ∗

0 ∗ )

},D={ ( ∗ 0

0 ∗ )

},U={ (

1 ∗ 0 1

) }.

Then,Uis normal subgroup ofB,U D=B,andU∩D={1}^5–8. The groupBis called theBorel subgroupofG andB=U D,a semidirect product. To find the presentation, we need the following lemmas.

Lemma 1 (5.4.5)⁹LetGbe a group of finite orderNin which every Sylow subgroups is cyclic. ThenGis generated by two elementsAandCwith defining relations

A^m=Cⁿ=I, CAC⁻¹=A^′,N=nm ((r−1)n,m) =1,rⁿ=1( mod m) .

Lemma 2 LetBbe the Borel subgroup ofSL(2,Fp).Then every Sylow subgroups ofBis cyclic.

Proof

Firstly, the subgroupUis the Sylowp−subgroupBand this group is generated by (

1 1 0 1

) .

In caseℓ|p−1.The Sylowl−subgroups is the subgroups ofD.SinceDis cyclic, these groups are also cyclic.

Proposition 3 LetBbe a Borel subgroup ofSL(2,Fp).Then B=

⟨

T,ya|T^p=I,y_a^p−1=I,yaTy⁻_a¹=T^a2

⟩ ,

whereais a generator of(Z/p)^∗.Moreover,

H₁(B)∼=Z/(p−1).

Proof.We begin with the first observation (

1 x 0 1

)(

a 0 0 a⁻¹

)

= (

a xa⁻¹ 0 a⁻¹

)

wherex∈Z/panda∈(Z/p)^∗.

Since(Z/p)^∗is a group, an elementx^a−1runs through all a set{0,1, . . . ,p−1}whenxruns throughZ/p.By setT=

( 1 1 0 1

)

,we have

( 1 x 0 1

)

=T^x.

Next, byB∼=Z/p×Z/(p−1), we get

yaxy⁻_a¹=x^a2, whereya=

( a 0 0 a⁻¹

)

. Now letabe a generator element of(Z/p)^∗.Sinceyais the diagonal matrix, we get

⟨ya⟩= {(

b 0 0 b⁻¹

)

|b∈(Z/p)^∗ }

.

Therefore,

B=

⟨

T,ya|T^p=I,y^p_a⁻¹=I,yaTy⁻_a¹=T^a2

⟩

these relations are maximum by Lemma 1 (Notice that we can apply Lemma 1 sinceBsatisfies Lemma 2).

Hence, by (1) we get

H1(B) =B/⟨B,B⟩=

⟨

T,ya|T^p=I,yap−1

=I,T^a2−1=I

⟩ .

Assumea²−1≡k(modp).Then(k,p) =1,now there arer,tsuch that kr + pt=1,

hence,

This implies thatT=1.Thus we getH1(B)∼=Z/(p−1).

The finite groupGisperiodicand of periodn>0if and only ifHⁱ(G,Z)∼=Hⁱ⁺ⁿ(G,Z)fori≥1.Arcoding to Thomas¹⁰the groupGis a periodic group if and only if ap−Sylow subgroup is either cyclic or generalised quaternion/binary dihedral (ifp=2). From Lemma 2, we getBis the periodic groups but the period is un- known.

Thep−periodof groupGis the period ofHⁿ(G,Z)_(p).Whenpis odd, it is easy to calculate thep−period.

Lemma 4 (¹⁰)LetNpbe the normalizer of thep−Sylow subgroup ofGandZpits centralizer. Then thep−period is2[

Np:Zp

].

Proposition 5 Thep−period ofBisp−1.

Proof.The Sylowp−subgroup ofBis the subgroupU.By the Proposition 3, its normalizer is the whole group Band its centralizer is generated by−I,U.By Lemma 5, we get thep−period is

2(p(p−1)/2p) =p−1,

since|N p|=p(p−1)and|Z p|=2p.

THE INTEGRAL HOMOLOGY OF BOREL SUBGROUPS B OF SL(2, Z /P)

In this section we will give a detailed computation of the Borel subgroup ofSL(2,F p).In order to compute the integral homology ofB, we decompose it intoℓ-primary parts

Hn(B,Z) =⊕ℓ|order(B)Hn(B,Z)_(ℓ)=Hn(B,Z)_(p)⊕(

⊕q̸=p

)Hn(B,Z)_(q).

To compute the first summand, we are concerned with the ring cohomology ofp−groups. In cohomology there is a cup product induced from the diagonal map and the Kunneth formula. In particular, with (untwisted) field coefficientsF,this gives a pairing

∑

i,j

Hⁱ(G,F)⊗_FH^j(G,F)→H^i+j(G,F)

makingH^∗(G,F):=∑iHⁱ(G,F)into an associated commutative ring with unit. The following lemma gives us the structure of ring cohomology ofp−groups.

Lemma 6 (¹)Letpbe an odd prime, thenH^∗( Z/p,Fp

)=E(v1)⊗Fp[b2],the tensor product of a polynomial algebra on a two dimensional generator and an extorior algebra on a1−dimensional generator.

Theorem 7

Hk(B,Z)(p)= {

0 otherwise

Z/p if k≡0(p−2) .

Proof. LetBbe a Borel subgroup ofSL(2,Fp).Then thep−Sylow subgroup isU∼=Z/p,this group is also normal inB,so we have

H^∗(B,Fp) = H^∗(Z/p,Fp)^F∗p.

Using ring structure from Lemma 6 H^∗(

Z/p,Fp

)_Z/p

=(

E(x)⊗Fp[y])_F∗ p

withyin cohomological degree 2 andxin cohomological degree 1. The action is multiplicative and determined byax:=a²x,ay:=a²y(

ais generator ofZ^∗p

).

The elements ofE(x)⊗Fp[y]only have the forms∑^i=pi=0⁻¹aiyⁱand∑^i=pi=0⁻¹aixyⁱcausex²=0( ai∈Fp

). Under the above action, we have

a (_i=p−1

i=0

∑

a_iyⁱ )

=

i=p−1 i=0

∑

a_i (

a²y )i

,and a

(_i=p−1

i=0

∑

a_ixyⁱ )

=

i=p−1 i=0

∑

a_i (

a²x )(

a²y )i

.

By Fermats little Theorem, the invariant must be generated byy^(p−1)/2andy^(p−3)/2x.It implies thatZ/pjust appear in the positionk=0mod(p−2)ork=0mod(p−1).Therefore, fork̸=0

H^k(B,Z)(p)=















0otherwise Z/p i f k≡0(p−2)

Z/p i f k≡0(p−1)

.

Now using the Dual coefficient theorem for group cohomology, we obtain

H_k(B,Z)(p)= {

0 otherwise

Z/p ifk≡0(p−2) .

To compute the rest summands, we use Lyndon-Hochschild-Serre cohomology spectral sequence with coeffi- cientZ[1/p]as follows

E²_p,q=Hp

(D,Hq(U,Z[1/p])⇒Hp+q(B,Z[1/p]).

Lemma 8GivenGis a finite group and the ringZ[1/p]as a trivialG−module. Then forn>0,Hn(G,Z[1/p])∼=

⊕q̸=pHn(G,Z)(q). In other words, the coefficientZ[1/p]kills thep−primary part in the integral homology ofG.

Proof.Using Universal Coefficient Theorem,

Obviously,Tor(H_n−1(G,Z),Z[1/p]) =0sinceZ[1/p]is torsion-free. Also tensoring withZ[1/p]kills the p−primary part ofHn−1(B,Z)since ifp^rx=0then

x⊗y=x⊗ (

p^r y p^r

)

=p^rx⊗ y p^r =0.

Moreover, ifq^rx=0for someqprime topthen there exista,b∈Zsuch thataq^r+bp^k=1. Thus x⊗m

p^k =x (

aq^r+bp^k )× m

p^k =bm⊗1.

Therefore,Hn(G,Z[1/p])∼=⊕q̸=pHn(G,Z)_(q).

Now consider the ringZ[1/p]as a trivialB−module. Then the ringZ[1/p]can be considered as a trivial U−module (a trivialT−module). Thus, the Lemma 8 gives us the following theorem.

Theorem 9Ht+s(B,Z[1/p]) =E_t,s² =

{ Z[1/p] ifs=t=0 Zp−1 if s=0and todd

0 otherwise

.

Proof.Hs(U,Z[1/p]) =0for alls>0andH₀(U,Z[1/p])∼=Z[1/p].Since onlyH₀(U,Z[1/p])is non-zero and the groupTacts trivially onH0(U,Z[1/p])we haveE_t,0² =Ht

(F^∗p,Z[1/p])∼=Z/(p−1)fortodd. Obviously, E²=E^∞⇒Hn(B,Z[1/p]).

Lemma 8 also gives us that

Hn(B,Z[1/p]) =⊕q̸=pHn(B,Z)_(q).

Hence,Hn(B,Z) =Hn(B,Z)(p)⊕Hn(B,Z[1/p]). In conclusion, one gets the following theorem.

Theorem 10 Forp≥5.Then

H_n(B,Z) =

{ Z/(p−1) if n is odd and n ̸≡(p−2) Z/(p−1)⊕Z/p ifn≡0(p−2)

0 otherwise

.

COMPETING INTERESTS

Hn(G,Z[1/p])=Hn(G,Z) ⊗_ZG Z[1/p] ⊕Tor_ZG(H_n−1(G,Z),Z[1/p]).

AUTHOR CONTRIBUTION

Vo Quoc Bao have contributed the presentation and the periodicity of Borel subgroup of SL(2,Fp) and have written the manuscript. Bui Anh Tuan have contributed the integral homology of Borel subgroups B of SL(2,Fp) and revising the manuscript.

ACKNOWLEDGMENTS

The authors gratefully acknowledges the many helpful suggestions of Hans–Werner Henn and Lionel Schwartz.

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