Groups Algorithms Programming / Computational...

What is group homology? What is gap? A couple of algorithms

Groups Algorithms Programming /Computational Homological Group Theory

Santiago de Compostela Summer School, July 2007

Graham Ellis & Marc Roderhttp://www.maths.nuigalway.ie


Lecture 1:

What is group homology?What is gap?A couple of basic algorithms

Lecture 2:

Integral cohomology of finite/infinite isometry groups andArtin groupsPolytopes, polymake and gap

Lecture 3:

Mod p cohomology of finite groupsLinear algebra computations with gap

Lecture 4:

Integral cohomology of finite groupsProgramming spectral sequences in gap


Tutorial 1:

Permutation groupsFinitely presented groupsLists and basic programming

Tutorial 2:

gap attributes and propertiesMore on permutation groups and finitely presented groupsFinite fields and matrix algebra

Tutorial 3:

Coset enumerationRecords in gap

Group homology and poincare series

Tutorial 4:

Cohomology of group extensionsCohomology of crystallographic groupsLie algebra homology


Heinz Hopf, “Fundamentalgruppe und zweiteBettische Gruppe”, Commentarii Mathematicii Helvetici, 1941:

Theorem

Let B be a path-connected space with π2(B) = 0. Then

H1(B , Z) = π1(B)ab, H2(B , Z) = R∩[F ,F ][R,F ] where F is any free

group with normal subgroup R and isomorphism F/R ∼= π1(B).


Definition (topological version)

The homology of a group G is defined to be the homology of theorbit space B = X/G where X is any contractible space on whichG acts freely.


Example

G = 〈P ,Q | PQ = QP〉 acts on X = R2

P · (x , y) = (x + 1, y)Q · (x , y) = (x , y + 1)

X

PQ

B = X/G


Example

G = 〈P ,Q | PQ = QP〉 acts on X = R2

P · (x , y) = (x + 1, y)Q · (x , y) = (x , y + 1)

X

PQ

B = X/G

Hn(G , Z) = Hn(B, Z) =

0, n = 0Z ⊕ Z, n = 1Z, n = 20, n ≥ 3


The cellular chain complex

C∗(X ) : · · · → C3(X )d3→ C2(X )

d2→ C1(X )d1→ C0(X )

is a free ZG-resolution:

(Freeness) each Cn(X ) is a free ZG -module

(Exactness) Image(dn+1) = Kernel(dn) for n ≥ 1

(Augmentation) Cokernel(d1) is the trivial ZG -module Z

Definition (algebraic version)

For any free ZG -resolution C∗ and any ZG -module A, set

Hn(G ,A) =ker(Cn ⊗ZA A −→ Cn−1 ⊗ZA A)

image(Cn+1 ⊗ZA A −→ Cn ⊗ZA A)


GAP is a free, open and extensible software package forcomputation in discrete abstract algebra


GAP stands for Groups Algorithms Programming


Development began in 1985 under leadership of Joachim Neubuserat Aachen. Version 2.4 released in 1988.

Version 4.1 released in 1999 from St Andrews.

Current version is 4.4 .


GAP contains

a kernel written in C:

rational arithmetic, finite fields, linear algebra, permutationsand words, lists, records, . . .interpreter for the GAP programming language.a few functions for handling external files and executingexternal programs.a few functions for testing, debugging and timing algorithms.


GAP contains

a large library of algorithms implemented in the GAPprogramming language.


GAP contains

data libraries:

all groups of order at most 2000 except order 1024Brauer character tablestables of marks.


GAP contains

many independent GAP packages of algorithms


GAP contains

documentation; discussion forum; user support team


GAP contains

a kernel written in C:

rational arithmetic, finite fields, linear algebra, permutationsand words, lists, records, . . .interpreter for the GAP programming language.a few functions for handling external files and executingexternal programs.a few functions for testing, debugging and timing algorithms.

a large library of algorithms implemented in the GAPprogramming language.

data libraries:

all groups of order at most 2000 except order 1024Brauer character tablestables of marks.

many independent GAP packages of algorithms

documentation; discussion forum; user support team


Who uses GAP?

Students taking an undergraduate/postgraduate algebracourse

Researchers

Others?


gap> Order( Group( [ (1,2), (1,2,3,4,5,6,7,8,9,10) ] )

3628800

gap>


gap> Order( Group( [ (1,2), (1,2,3,4,5,6,7,8,9,10) ] )

3628800

gap>

gap> TimeToString(time);

‘‘0.002 sec.’’

gap>


gap> Order( Group( [ (1,2), (1,2,3,4,5,6,7,8,9,10) ] )

3628800

gap>

gap> Factorial(10);

3628800

gap>


gap> for G in AllSmallGroups(32) do

> if DerivedSubgroup(G) = Center(G) then

> Print(StructureDescription(G),‘‘\n’’);

> fi;

> od;


gap> for G in AllSmallGroups(32) do

> if DerivedSubgroup(G) = Center(G) then

> Print(StructureDescription(G),‘‘\n’’);

> fi;

> od;

(C2 x C2 x C2 x C2) : C2

(C4 x C2 x C2) : C2

(C2 x Q8) : C2

(C4 x C2 x C2) : C2

(C4 x C4) : C2

(C2 x C2) . (C2 x C2 x C2)

(C4 x C4) : C2

(C4 x C4) : C2

C4 : Q8

(C2 x D8) : C2

(C2 x Q8) : C2

gap>


gap> LoadPackage(‘‘HAP’’);



gap> S:=SylowSubgroup(SL(2,Integers mod 5^3),5)

gap> K5:=MaximalSubgroups(S)[2];

gap> GroupHomology(K5,3);

[ 5, 5, 5, 5, 5, 5, 125 ]



gap> S:=SylowSubgroup(SL(2,Integers mod 5^3),5)

gap> K5:=MaximalSubgroups(S)[2];

gap> GroupHomology(K5,3);

[ 5, 5, 5, 5, 5, 5, 125 ]

K5 = ker(SL2(Z53) → SL2(Z5)) has order 56 = 15, 625 .

W. Browder & J. Pakianathan [ Trans. American Math. Soc.,1999] first showed H3(Kp, Z) has exponent p3 for odd p .

Pakianathan: Hn(Kp, Z) has exponent ≤ p2 for all but finitelymany n and so Kp is a counter-example to a conjecture ofA. Adem.


gap> G:=Group( [ (1,2,3,4,5,6,7,8,9,10,11,12,13,

> 14,15,16,17,18,19,20,21,22,23),

> (3,17,10,7,9)(4,13,14,19,5)

> (8,18,11,12,23)(15,20,22,21,16) ]);;

gap> GroupHomology(G,1);

[ ]


[ ]


[ ]


gap> G:=Group( [ (1,2,3,4,5,6,7,8,9,10,11,12,13,

> 14,15,16,17,18,19,20,21,22,23),

> (3,17,10,7,9)(4,13,14,19,5)

> (8,18,11,12,23)(15,20,22,21,16) ]);;


[ ]


[ ]


[ ]

J. Milgram [J. Group Theory, 2000] first showed thatHn(M23, Z) = 0 for 1 ≤ n ≤ 3, thus disproving a conjecture madeby J.-L. Loday in the 1970s.


gap> G:=MathieuGroup(11);;

gap> Order(G);

7920

gap> PoincareSeriesPrimePart(G,2,15);

(x^4-x^3+x^2-x+1)/(x^6-x^5+x^4-2*x^3+x^2-x+1)


gap> G:=MathieuGroup(11);;

gap> Order(G);

7920

gap> PoincareSeriesPrimePart(G,2,15);

(x^4-x^3+x^2-x+1)/(x^6-x^5+x^4-2*x^3+x^2-x+1)

The Poincare series for H∗(M11, Z2) was first calculated (and

proved) in [P.Webb, “A local method in group cohomology”,Comm. Math. Helv. 62 (1987) 135-167].


Algorithm 1

Given a generating set Y for a permutation group G , we obtain afinite presentation for G by using the Cayley graph Γ = Γ(G ,Y ):

one vertex for each g ∈ G ,

one edge [g , gy ] for each g ∈ G , y ∈ Y .


Algorithm 1




Any basis for H1(Γ, Z) corresponds to a set of relations R suchthat G = 〈Y : R〉 .


Algorithm 1





Redundant relators in R can be discarded.


Algorithm 1





Redundant relators in R can be discarded.

Exercise: Find a presentation for G = 〈 (1, 2), (1, 2, 3) 〉.


gap> Y:= [(1,2), (2,3), (3,4) ] ;;

gap> G:=Group( Y );;

gap> CayleyGraphDisplay(G,Y);


gap> Y:= [(1,2), (2,3), (3,4) ] ;;

gap> G:=Group( Y );;

gap> CayleyGraphDisplay(G,Y);

G = 〈x , y , z : x2 = y2 = z2 = (xy)3 = (yz)3 = (xz)2 = 1〉


Procedure 2

There is no algorithm for deciding if a presentation 〈Y : R〉presents a finite group G .


Procedure 2

The Todd-Coxeter procedure is often successful in producing apermutation representation for G when G is finite. The idea is:

Assign permutations π(y) to each of the generators y ∈ Y in sucha way that the permutations satisfy only the relations π(r) forr ∈ R .


Procedure 2

The Todd-Coxeter procedure is often successful in producing apermutation representation for G when G is finite. The idea is:

Assign permutations π(y) to each of the generators y ∈ Y in sucha way that the permutations satisfy only the relations π(r) forr ∈ R .

Exercise: Find the order, and a permutation representation, ofG = 〈 x , y : yxyx−1 = 1, xyxy−1 = 1 〉.


Algorithm 3

Given a set Z = {z1, . . . , zk} ≤ Sn of permutations, the order ofthe group G = 〈Z 〉 can be found using:

|G | = |nG | × |Gn|

(where nG = {ng : g ∈ G}, Gn = {g ∈ G : ng = n}).


Algorithm 3


|G | = |nG | × |Gn|


Gn is generated by

Zn = {(tz)−1

tz : t ∈ T , z ∈ Z}

where T is a transversal for Gn in G , and the coset gGn isrepresented by g ∈ T .


Algorithm 3


|G | = |nG | × |Gn|


Gn is generated by

Zn = {(tz)−1

tz : t ∈ T , z ∈ Z}

where T is a transversal for Gn in G , and the coset gGn isrepresented by g ∈ T .

Exercise: Find the order of the permutation groupG = 〈 (1, 2), (1, 3, 5) 〉.

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