the cohomology of a finite matrix quotient group

THE COHOMOLOGY OF AFINITE MATRIX QUOTIENT GROUP byBRIAN PASKOA.A.S., Milwaukee Area Technical College, 1996 B.S., Marquette University, 1998M.S., Kansas State University, 2001 AN ABSTRAC

THE COHOMOLOGY OF A

FINITE MATRIX QUOTIENT GROUP

byBRIAN PASKOA.A.S., Milwaukee Area Technical College, 1996

B.S., Marquette University, 1998M.S., Kansas State University, 2001

AN ABSTRACT OF A DISSERTATION

submitted in partial fulfillment of the

requirements for the degree

DOCTOR OF PHILOSOPHYDepartment of MathematicsCollege of Arts and Sciences

KANSAS STATE UNIVERSITY

Manhattan, Kansas

2006

In this work, we find the module structure of the cohomology of the group of four by fourupper triangular matrices (with ones on the diagonal) with entries from the field on threeelements modulo its center Some of the relations amongst the generators for the cohomologyring are also given This cohomology is found by considering a certain split extension Weshow that the associated Lyndon-Hochschild-Serre spectral sequence collapses at the secondpage by illustrating a set of generators for the cohomology ring from generating elements ofthe second page We also consider two other extensions using more traditional techniques

In the first we introduce some new results giving degree four and five differentials in spectralsequences associated to extensions of a general class of groups and apply these to both theextensions

THE COHOMOLOGY OF A

FINITE MATRIX QUOTIENT GROUP

byBRIAN PASKOA.A.S., Milwaukee Area Technical College, 1996

B.S., Marquette University, 1998M.S., Kansas State University, 2001

A DISSERTATIONsubmitted in partial fulfillment of the

requirements for the degree

DOCTOR OF PHILOSOPHYDepartment of MathematicsCollege of Arts and Sciences

KANSAS STATE UNIVERSITY

Manhattan, Kansas

2006

Approved by:

Major ProfessorJohn Maginnis

UMI Number: 3229970

3229970 2006

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by ProQuest Information and Learning Company

In this work, we find the module structure of the cohomology of the group of four by fourupper triangular matrices (with ones on the diagonal) with entries from the field on threeelements modulo its center Some of the relations amongst the generators for the cohomologyring are also given This cohomology is found by considering a certain split extension Weshow that the associated Lyndon-Hochschild-Serre spectral sequence collapses at the secondpage by illustrating a set of generators for the cohomology ring from generating elements ofthe second page We also consider two other extensions using more traditional techniques

In the first we introduce some new results giving degree four and five differentials in spectralsequences associated to extensions of a general class of groups and apply these to both theextensions

List of Tables vi

Notation vii

Acknowledgements viii

1 Introduction 1 1.1 Background 1

1.2 History 7

1.3 Preliminaries 9

1.4 Maps from subgroups 11

1.5 Representation Theory 14

1.6 Massey products 16

1.7 G27 18

List of Tables

1.1 Restrictions in G27: 20

1.2 Transfers from < c >' Z/3Z in G27: 21

2.1 Restrictions in H 32

2.1 Restrictions in H (continued) 33

2.2 Java Output 35

5.1 Restrictions in U 57

A.2 Transfers in H 64

GLn(R) invertible n × n matrices with entries from the ring R

U Tn(R) n × n upper triangular matrices with entries from the ring R

H∗(G; M ) Cohomology of the finite group G with coefficients in

the G-module M

< x, y, z > Massey product of classes x, y, z

NG

F[α1, α2] Algebra over the field F generated by α1 and α2

F[α1, α2](x1, x2) F[α1, α2]-module with basis {x1, x2}

I would like to thank my advisor, Dr John Maginnis for his encouragement, support andkind patience throughout my PhD work I would also like to thank Messrs M Beswick, S

H Kim and S Koshkin for their friendship and constant source of philosophical discourse

An especial thanks to Mr D Adongo both for his familial friendship and for writing theJava program used to perform some calculations in this work

to solve a problem using one approach one translates it into the language of the other areaand then (hopefully) decides the question using known results.

How the group structure is reflected in the cohomology of the group is still largely known Some important information however is For example, there is a 1-1 correspondencebetween splittings of the split extension (short exact sequence) 1 → N ,→ G Q → 1 andqelements in H1(G; M ) When the normal subgroup N is abelian, we have a another 1-1correspondence between the set of equivalent extensions of the above form and H2(G; M ).That is, we can up to isomorphism, decide how many groups G fit into this extension once

un-we fix N and Q One may suppose that that cohomology is useful in distinguishing groups.Alas, different groups can have the same cohomology The converse, however, is true That

is, groups with differing cohomology cannot be isomorphic There are also connections tween the mod-p cohomology ring of a group and the modular representations of the group.Indeed, the cohomology groups of GL4(F3) give universal characteristic classes for modulargroup representations

be-The main tool in group cohomology is the spectral sequence This device provides acomputational means of finding the cohomology of a group Generally, a spectral sequence

is a sequence of objects called pages that ”converges” to the cohomology of the group G Apage in a spectral sequence is a 1st quadrant array of modules along with a multiplicationmaking the page a ring The rth page also has a differential, dr, defined on it That is,

a map of bidegree {r, −r + 1} which is a derivation with respect to the multiplication.Derivation here means that dr(ab) = dr(a)b + (−1)|a|adr(b), where if a is in the {i, j}-position (written a ∈ Ei,j

r ), then |a| = i + j ; and {r, −r + 1}-bigraded means that if

a ∈ Ei,j

r , then dr(a) ∈ Ei+r,j−r+1

r Given the Er − page define the {i, j} entry of the

Er+1− page as Kerdr|Ei,j

r modulo Imdr|Ei−r,j+r−1

r That is, Er+1 is the homology of Er.Consider again the extension 1 → N ,→ G  Q → 1 A well known theorem says thatqthe second page of the Lyndon-Hochschild-Serre spectral sequence is given as the cohomology

of the quotient with coefficients being the cohomology of the normal subgroup Symbolically,

E2i,j ' Hi(Q; Hj(N ; M )) With this formula in hand, how to proceed is clear Chooseextensions with sub- and quotient groups having cohomologies that we understand well andfrom this knowledge determine the cohomology of G We obtain in the words of McCleary:

”E2i,j ' ”something computable” converging to H∗, something desireable” ([8, Pg 6])

Of particular interest to algebraists is H∗(GL4(F3); F3), the group of four by four vertable matrices over the field with three elements One way to find this cohomology

in-is to first find H∗(U T4(F3); F3), the upper triangulars with ones on the diagonal, andthen since H∗(GL4(F3); F3) injects into H∗(U T4(F3); F3) we may work backward to de-rive H∗(GL4(F3); F3) Hereafter, let H denote the quotient group U T4(F3) modulo itscenter The center is a copy of Z/3Z generated by I4+ a1,4 This dissertation will addressthis group of order 35

The outline of this thesis is as follows: the current chapter gives background on thequestion of interest, and some standard definitions and results used in the later chaptersincluding a more rigorous development of the ideas above Chapter 2, our main result,concerns the spectral sequence associated to the extension (Z/3Z)4 −→ H −→ Z/3Z

We decompose H∗((Z/3Z)4; F3) as a sum of indecomposable F3[Z/3Z]-modules to decidethe F3[Q]-module structure of the coefficients, H∗((Z/3Z)4; F3) From this data we mayconstruct the E2− page of the associated Lyndon-Hochschild-Serre spectral sequence Ac-cording to theorem 2.7, E2 = E∞ which is proved by showing that a minimal set of ringgenerators of the E2 − page represents a minimal set of generators for E∞ We definethe ring generators of H∗(H; F3) as Evens norms, Massey products and transfers and theirBocksteins A complete set of multiplicative relations in H∗(H; F3) is not given although,

a partial list is presented in appendix A

Chapter 3 considers the central extension (Z/3Z)2 −→ H −→ (Z/3Z)3 Standardtheorems give some differentials in the associated spectral sequence and some new results

providing some d4 and d5 differentials are produced (on pages 42–43) These are of interestbecause of the general lack of theorems giving differentials in spectral sequences Thecalculation of H∗(H; F3) is not completed in this section although some comparison ismade with the results of chapter 2.

Chapter 4 considers the central extension Z/3Z −→ H −→ G27× Z/3Z Again, dard theorems as well as the results from chapter 3 give some differentials We also introduceand apply Leary’s circle method to this extension and observe some limitations of this tech-nique

stan-The final chapter lists some immediate and distant desired results that this work points

to Some initial consideration of H∗(U T4(F3); F3) is made including a theorem with the d2and d3 differentials in the spectral sequence associated to the central extension

F3[G]-module maps, and ∂n−1 ◦ ∂n = 0 Apply the contravariant functor HomF3[G]( , M )dimension-wise to obtain a cochain complex, (X0, δ)

· · · ← Xn+1 δ n

← Xn δ n−1

← Xn−1· · · ← X1 ← X0

We then define the cohomology of G with coefficients in M as follows:

Hn(G; M ) = Ker(δn) / Im(δn−1) The reader may recognize this as ExtnF3[G](F3, M ), or asthe simplicial cochain complex of the topological space K(G, 1)

As an example, let G = Z/pZ and let M be the trivial Fp[G]-module Fp Suppose

G =< a > A projective resolution, X → Fp is Xi = Fp[G]ei for i ≥ 0, and

The cohomology of a group G admits a graded ring structure when the coefficients are

a ring Graded here means that αβ = (−1)|α||β|βα , where |z| denotes the degree of theclass z ∈ H∗(G; Fp) Definitions and proofs of various properties of this cup product may

be found in standard references (e.g [5], [7]) What concerns us for the moment is the factthat we may use this product to write cohomologies as algebras Thus, for example, wemay write H∗(Z/pZ; Fp) as the tensor product of a polynomial algebra on one degree twogenerator α and an exterior algebra on one degree one generator x: Fp[α] ⊗ ∧Fp(x)

We will often write cohomologies as modules over a polynomial subalgebra and refer tothis description as an additive description For example, H∗(Z/pZ; Fp) ' Fp[α] ⊗ ∧Fp(x)

we write as Fp[α](1, x); and, H∗((Z/pZ)2; Fp) ' Fp[α, β] ⊗ ∧Fp(x, y) as Fp[α, β](1, x, y, xy),

where x, y are degree one generators and α, β are degree two generators.

Continuing the example above, by the universal coefficient theorem, H∗((Z/pZ)m; Fp) '

F3[α1, αm](1, x1, xm, xixj, , xixjxk, x1x2 xm) 

1.2 History

Let U Tn(Fp) denote the n by n upper triangular matrices (with ones on the diagonal)with entries from the field on p elements, Fp Let GLn(Fp) denotes the invertible n by nmatrices over Fp

The cohomology of the two by two upper triangulars, U T2(Fp) is well known Indeed, thisgroup is isomorphic to the cyclic group of order p Quillen [16] found H∗(U T3(F2); F2) Inthis work he introduced many new and powerful techniques to approach the cohomology of agroup Indeed, this work largely marked the beginning of modern group cohomology Lewislater [11] decided the ring structure of H∗(U T3(F3); Z) (Note: this group is isomorphic tothe extra special group of order 27 and exponent 3.) In the 1980’s and 90’s, several authorsgave arguments to find H∗(U T3(F3); F3) This cohomology appeared as a subalgebra ofwork by Mimh [15] who gave the ring structure, although, Milgram and Tezuka report thathis answer was incorrect

In [10], Leary gives the complete ring structure of H∗(U T3(F3); F3) including a full set ofmultiplicative relations He introduced his ’circle method’ in this, his PhD thesis We shalladdress this method later Leary revisited H∗(U T3(F3); F3) in [9] This time he attackedthe problem quite differently He again considered a central extension but rather thanapplying the circle method, he used the known results from his previous work to find the d2and d3 differentials The Kudo and Serre transgression theorems gave some d5 differentialsand additionally, the author was able to produce new results giving the d4 differentialcompletely We shall see generalizations of these results in Chapter 3 Benson and Carlson[4] published a summary of the work to date (1991) on H∗(U T3(F3); F3) and more generally,the extraspecial p groups including a algebraic determination of H∗(U T3(F3); F3)

Published the same year as [9], Milgram and Tezuka [14] obtain the ring H∗(U T3(F3); F3)

by using Lewis’ result They used the fact that the short exact sequence of coefficients

Z→ Z → Z/pZ induces a long exact sequence in cohomology:p

as writting this algebra in ’normal form’) while Leary did not Of course, given a set ofgenerators and relations, one can construct an additive description, however, the converse

is not true

Siegel [17] introduced means of constructing the E2− page of a spectral sequence ciated to a certain extension using representation theory, one which we shall make use of toobtain our principle result He obtained the additive structure but stopped short of findingthe relations or giving an additive description

asso-This discussion leads to the question: what is H∗(U T4(F3); F3) ? Some comments on thecohomology of U T4(F3) are given in chapter 5 but, this tract will mainly concern H∗(H; F3)

1.3 Preliminaries

We record some basic facts about matrix groups Hereafter, we let ai,j = In+ ei,j, where

ei,j is the n × n matrix with 1 in the i,j-entry and zeroes elsewhere

The Sylow-p subgroup of GLn(Fp) is the n × n upper triangular matrices with 1’s onthe diagonal, U Tn(Fp) The center of U Tn(Fp) is a copy of Z/pZ generated by a1,n Thecenter of the quotient of U Tn(Fp) by its center is a copy of Z/pZ × Z/pZ generated by

In general, define Hk= {A ∈ U Tn(Fp)|aij = 0 for 0 < j − i < k}

Proposition 1.1 The collection {sai,i+1}n−1i=1 generates U Tn(Fp)

Proof: We will only concern ourselves with the cases n = 3, 4 for which the result iseasy to see by calculation The more general proof is given in [18] which is just the obvious

Let H be U T4(F3) modulo its center, H = U T4(F3)/ < a1,4 > Make the followingidentifications: ai denotes ai,i+1, a2i+2 denotes ai,i+2 and a6 denotes a1,4 We will use afew convenient abuses of notation A ’subgroup generated by ai’ will mean the subgroup

< aij > as appropriate Also, we will write elements in the group H as ai rather than ai;and, elements of quotient groups of H as ai instead of ai, etc It should be clear from the

context which group and which elements we’re referring to The group H is generated by

a1, a2, a3 Indeed, [a1, a2] = a4 and [a2, a3] = a5, and [a4, a5] = e H contains many copies

of U T3(F3) The two copies < a4, a2, a1 > and < a5, a3, a2 > will be used regularly in themain body of this text We will call these subgroups Gh

27 and Gl

1.4 Maps from subgroups

This section contains material familiar to the area of study It is included here to for thesake of the expository nature of this work All the material in this section may be found

in many references, among them [3], [5], [7] We will make extensive use of these ideasthroughout this text explicitly or otherwise The transfer and Evens norm we present inthe restricted case of finite groups and Fp coefficients since this is all that will be required

in this work The interested reader can find the more general statements and applications

in the texts cited above

In later chapters we will use 1.6 to show that certain elements in spectral sequences live

to the E∞− page We will be able to define non-zero classes in the cohomology of the group

H as Evens norms or transfers of classes in a particular subgroup In addition, in Appendix

A, Proposition 1.3 will allow us to find relations amongst the ring generators of a group’scohomology

We will not elaborate on the obvious maps ResG

N: H∗(G; M ) → H∗(G; M ) induced bythe inclusion N ,→ G; and InfG

N: H∗(G/N ; MN) → H∗(G; M ) induced by the canonicalquotient map G  G/N when N
G The Bockstein map is the connecting homomor-qphism β : H∗(G; Z/pZ) → H∗+1(G; Z/pZ) in the long exact cohomology sequence induced

by the sequence of coefficients Z/pZ−→ Z/pp 2

Z −→ Z/pZ

Transfer

Let D be a set of coset representatives for a subgroup of finite index, H , in G Let

X → Fp be a resolution of Fp over Fp[G] Note that by restriction it is also a Fpresolution Suppose the cochain f represents α ∈ H∗(H; Fp) At the cochain level, the

[H]-transfer is given as the map T : HomFp[H](X, Fp) → HomFp[G](X, Fp) by

Definition 1.2 The transfer is the map TrGH : H∗(H; Fp) → H∗(G; Fp) induced by T

Proposition 1.3 Let H, K < G with [G : H] < ∞ and let D be a set of coset tatives for the double cosets HgK For α ∈ H∗(G; M ),

Let X denote G/H and Y denote H Let S = {s ∈ S(X×Y )|s(x, y) = (s(x), y) ∀ (x, y) ∈

X × Y } The product of |X| copies of H , HX, is in 1:1-correspondence with the set{s ∈ S(X × Y )|s(x, y) = (x, hx(y)) ∀ (x, y) ∈ X × Y for some hx ∈ H} It is easy tosee that S ∩ HX consists only of the identity and that S is in the normalizer of HX inS(X × Y ) We then define the wreath product of X and Y , W = HX

o S and denote it by

W = SR H

We now define the map 1R α : H∗

(H; Fp) → H∗(S(G/H)R H; Fp) To do so we requireNakaoka’s theorem:

Theorem 1.4 H∗(S(G/H)R H; Fp) ' H∗(S; H∗(H; Fp)⊗G/H)

Note that H0(S; H∗(H; Fp)⊗G/H) consists of those classes in H∗(H; Fp)⊗G/H invariantunder the action of S For α ∈ Heven(H; Fp), the class

x∈DKxH is a double coset decomposition of G, then for all α ∈ H∗(H; Fp)

of even degree, ResGK(NG

H(α)) = Q

x∈D

NK K∩xHx −1(ResxHxK∩xHx−1 −1(cx−1(α))) When H
G, thisbecomes ResGK(NHG(α)) = Q

x∈D

cx−1 (α)



1.5 Representation Theory

In this section we gather some basic definitions and facts from modular representation theorythat will be used in this work The reader is advised to see [2] for additional details Inthis section, A is a finite-dimensional algebra with unit element over Fp, and U is a leftA-module that is finite-dimensional over Fp

Definition 1.7 The socle, socA(U ), is the largest semisimple (i.e direct sum of simples)submodule of U

Definition 1.8 The radical, radA(U ), is the smallest submodule of U with semisimplequotient

Alperin [2] states the following theorem The proof is omitted here since, in Alperin’swords, ” [it] presents no techniques we shall have any further use for” ([2, Pg 17]) We areinterested in a corollary of this result which places important conditions on the structure of

Proof: The trivial module is simple and by 1.9 the identity is the only conjugacy class

In chapter 2 we will use the usual minimal resolution for the group Q ' Z/pZ =< a >.Let S be a F3[Q]-module

Proposition 1.11 The socle and radical of S are given by:

i ) socF3[Q](S) = Ker(a − 1)

ii ) radF3[Q](S) = Im(a − 1)

Proof: i ) The socle socF3[Q](S) is a sum of simple F3[Q]-modules By 1.10 on thepreceding page, these simples are copies of the trivial module Fp thus these simples arecontained in Ker(a − 1) Now, Ker(a − 1) is a trivial module and is therefore semisimple.Since socF3[Q](S) is the largest semisimple submodule, Ker(a − 1) ⊂ socF3[Q](S)

ii ) It is well known that radF3[Q](S) = rad(F3[Q])S The subalgebra of F3[Q],rad(F3[Q]) is the collection of elements which annihilate every simple F3[Q]-module Thus

Proof: i ) We define soc2

F 3 [Q](S) = q−1(soc(S/socF3[Q](S)) where q : S → S/socF3[Q](S) isthe canonical quotient map Using the alternative description of socF3[Q](S) as the collection

of all u ∈ S such that rad(F3[Q])u = 0 and part ii of 1.11, the result is clear

ii ) Clear: rad2F

1.6 Massey products

The Massey triple product will be used throughout this text Let P∗ be the bar resolutionfor Fp over Fp[G] Suppose x, y, z ∈ H∗(G; Fp) with representative cocycles x, y, z in P∗satisfy xy and yz are both zero in H∗(G; Fp) Then, for some α ∈ Pdeg x+deg y−1 and

β ∈ Pdeg y+deg z−1, δ(α) = xy and δ(β) = yz

For x, y and z as above, we call the collection of cochains Aij = (aij) with

a11 = x, a22= y, a33 = z, a12 = α and a23 = β a defining system for the Massey product

of x, y and z The cocycle (−1)|α|αz − xβ is a related cocycle for the Massey product of x,

y and z

Definition 1.13 The Massey product < x, y, z > is defined to be the class represented

by all such related cocycles This operation is only well defined modulo xH|y||z|−1(G; Fp) +

H|x||y|−1(G; Fp)z since any cocycle representing a class in H∗(G; Fp) will have coboundaryzero We call this set the indeterminacy of the product < x, y, z >

Definition 1.14 Suppose the Massey product < x, x, x > is defined in H∗(G; Fp) If werestrict the defining system for this product Aij to aii= x and a12= a23, then, we call theresulting class a restricted Massey product and denote it < x >3

It is not hard to see that < x >3 is defined without indeterminacy This definition will

be useful in defining generating classes in cohomologies The proposition below will be usedimplicitly in later chapters

Proposition 1.15 [8, Theorem 1.4] If u ∈ H2m+1(G; Fp), then < u >p is defined as asingle class in H2mp+2(G; Fp) and < u >p= −β(Pmu)

Massey products satisfy the following well-known properties proofs may be found in [8]

or [10]: given u, v, w, x, y ∈ G such that all of the products are defined,

(a) Massey products are additive in each argument,

(b) < u, v, w > x + (−1)uu < v, w, x >≡ 0 mod uH∗x,

(c) (−1)wu < u, v, w > +(−1)uv < v, w, u > +(−1)vw < w, u, v >≡ 0

mod uH∗ + vH∗+ wH∗,

(d) < u, v, w > +(−1)|u||v|+|v||w|+|u||w| < w, v, u > mod uH∗+ H∗w

Proposition 1.16 Suppose x ∈ H1(G; F3), then − < x >3= β(x) =< x, x, x >

Proof: The first equality is from Proposition 1.15 and the second from [10] 2

Quadruple and higher Massey products are defined in various references [12], [8] Cleary [13] gives some interesting applications of Massey products to problems in Topology

Mc-In [12] May introduced further generalizations to matric Massey products A corollary ofhis work there is the following

Proposition 1.17 Every class in H∗(G; Fp) can be described as a matric Massey product

We will not have use of these generalizations in this work and so will not pursue these

1.7 G27

Let G27 be the extra-special group of order 27 and exponent 3 G27 ' U T3(F3) We maypresent G27 as < a, b, c|a3 = b3 = c3 = [a, c] = [b, c] = 1, [a, b] = c > As a ring, H∗(G27; F3)consists of nine generators which we label as follows:

αi = β(xi) =< xi, xi, xi >=< xi >3 according to 1.16 Take σ as < x1, x1, x2 > and τ

as < x1, x2, x2 > (see 1.18) Let µ = β(σ) and ν = β(τ ) The class γ we define as anEvens norm, γ = N (δ), where δ is a degree two generator in H∗(< a, c >; F3) which is theinflation of the degree two generator in H∗(< c >; F3)

For g ∈ G27, let g = ctbsau In the bar resolution for G27 consider the cochains:u[g] = u s[g] = s t[g] = t Note that u represents x1 and s represents x2

so that the equality makes sense

By computing the coboundary of u2 in the bar resolution for G27: δ(u2)[ctbsau|ct 0

bs0au0] =

u2{[ct 0

bs0au0] − [ct+t0+us0bs+s0au+u0] + [ctbsau]} = u02− (u + u0)2+ u2 = −uu0, we see that

δ(u2) = u1u2 (u1u2 represents the cohomology class x1 ∪ x1) Similarly, δ(s21) = s1s2 =

x2∪ x2 and δ(t1) = −u1s2 = −x1∪ x2 The system A with

a12 = u2, a23 = t, a11 = u, a22 = u and a33 = s is a defining system for < x1, x2, x2 > Arelated cocycle is u2

1s2− u1v2.The indeterminacy of σ is x1H1+H1x2 As a set, this indeterminacy is {x1x1, x1x2, x2x2}.But, x2

i = 0 since these are both degree one classes and x1x2 = 0 also Thus σ is definedwithout indeterminacy Setwise, the indeterminacy of σ is the same as τ and so is zero aswell Alternatively, we could quote a suitably modified version of a result of Kraines, 1.19 2

Proposition 1.19 [8, Lemma 20] For an odd prime p, if x, y, z ∈ Hodd(G; Fp) such that

< x, y, z > is defined, then < x, y, z > has no indeterminacy

in-H∗(G27; F3) to the Hij Restrict each class, z ∈ H∗(G27; F3) to see that ResG27

Proposition 1.21 G27 has no non-trivial essential cohomology That is, H∗(G27; F3)contains no non-trivial classes whose restriction to every subgroup is zero.

Proof: Directly from table 1.1 Or, by [14], H∗(G27; F3) is free and finitely generatedover a polynomial subalgebra (i.e is Cohen-Macaulay) and so, by [1] G27 does not containnon-trivial essential cohomology It should be noted that the additive description given onthe preceding page does is not describe this cohomology as Cohen-Macaulay 2

The following table of transfers is included in this section for completeness

Table 1.2: Transfers from < c >' Z/3Z in G27:

Chapter 2

A Non-central Split Extension

In this part, we consider the split extension

1 ) = ji; and, αi as the Bockstein of xi Similarly, we write H∗(Q; F3) =

F3[α2] ⊗ ∧F3(x2) and define x2 and α2 analogously The action of Q on H∗(N ; F3) is given

by a2(z5) = z5, a2(z4) = z4, a2(z3) = z5+ z3, a2(z1) = z4− z1 for zi = xi or αi

We follow the approach used by Siegel in [17] In this article Siegel gives a method forconstructing the E2 − page for a non-central extension where the quotient is cyclic Inoutline, he describes the E2 − page ' H∗(Q; H∗(N ; F3)) by decomposing H∗(H; F3) as adirect sum of indecomposable F3[Q]-modules

As in Siegel, for a F3[Q]-module M we have

Equation 2.2 is clear when we consider the usual minimal resolution for F3 over F3[Q],

aj2ei−1= (a2− 1)2ei−1 for i even,

We must determine the F3[Q]-module structure of H∗(N ; F3) There are, up to morphism, 3 indecomposable F3[Q]-modules (see [2, Pg 24 ff.]) We denote these by Ji,for 1 ≤ i ≤ 3, where Ji has dimension i Note that J1 = F3, J2 = F2

iso-3 and J3 = F3

3

J3 is projective, indeed J3 ' F3[Q] The notation Ji is used because the action of thegenerator a2 ∈ Q is given by a single Jordan block of size i We will write J2 =< x, y >and J3 =< x, y, z > if these are F3-bases cyclic with respect to a2− 1, that is (a2− 1)z = yand (a2− 1)y = x

Let W = F3[α1, α3](1, α4, α24)(1, α5, α25) and R = ∧F3(x1, x3, x4, x5)

Proposition 2.1 R = 5J1+ 4J2+ J3 and W = 2J1+ 2J2+ 3J3F3[α3] + 3J3F3[α1, α3].Proof: We show that R and W are as claimed by exhibiting bases for these modules.Let σ1 denote x1x4, τ2 = x3x5 and π = x1x5− x3x4

After collecting terms, S = 18J1⊕ 18J2⊕ 12J3⊕ 48J3F3[α3] ⊕ 48J3F3[α1, α3].

Proof: By [2, Lemma 5, pg 50], J2⊗ J2 ' J1⊕ J3 Of course, J1⊗ J ' J Since J3 is

We give bases for these modules When constructing a basis for J2 =< a, b > ⊗

< x, y >= J2, the resulting J1⊕ J3 can be given as:

< ay − bx > ⊕ < (a2− 1)2by, (a2− 1)by, by > Similarly,

In terms of our above labels, we may write:

Proof: Here as in Siegel (see 1.11, 1.12), socF3[Q](S) = Ker(a2− 1),

soc2

F 3 [Q](S) = Ker((a2 − 1)2), radF3[Q](S) = Im(a2 − 1) and rad2F3[Q](S) = Im((a2− 1)2) Inthe notation < x >, < x, y >, or < x, y, z > for the Ji given after Proposition 2.2 onpage 24, bases for the various submodules of interest are:

soc(J1) =< x > soc2(J1) =< x > rad(J1) = 0 rad2(J1) = 0

soc(J2) =< x > soc2(J2) =< x, y > rad(J2) =< x > rad2(J2) = 0

soc(J3) =< x > soc2(J3) =< x, y > rad(J3) =< x, y > rad2(J3) =< x >

F3[γ4, γ5] ⊗ socF3[Q] (S)

rad2

F3[Q] (S), r > 0 even

(2.3)

Write H∗(Q; F3) = F3[α2] ⊗ ∧F3(x2), x2, α2 defined as usual x2 generates E21,0 and

α2 generates E22,0 A product on the E2 − page is given as follows: suppose χ ∈ E2s,t isrepresented by x ∈ Hs((Z/3Z)4; F3) and ψ ∈ E2s0,t0 is represented by y ∈ Hs0((Z/3Z)4; F3)

Then, (−1)s0tχψ ∈ E2s+s0,t+t0 is represented by:

Theorem 2.5 The E2− page is additively isomorphic to:

α1θ2 may be rewritten by noticing that (a2− 1)2(α1x5α24α25) = λ(a2− 1)2(α24α25) − x3γ4(a2−1)2(α2

Corollary 2.6 E2 is generated as a ring by