Niemeyer Groups and Geometry Slides 2004

8/10/2019 Niemeyer Groups and Geometry Slides 2004

1/52

Groups and Geometry

Alice Niemeyer

Groups

Factor GroupsClassical Groups

Projective Spaces

Polar Spaces

Symplectic Groups

Generalised Quadrangles

Acknowledgements

Groups and Geometry

Alice Niemeyer

The University of Western Australia

September 2004
http://find/


2/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Simple Groups

In Chemistry compounds are made up ofelements.

Is there an analouge for finite groups?

YES! Simple groups are the building blocks.
http://find/


3/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Normal Subgroups

A subgroup Nof a group G is called normal

ifg1ngN for all gG and nN.

A group is called simpleif its only normalsubgroups are the group itself and the trivial

subgroup.
http://find/


4/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Factor Groups

The cosets of a normal subgroupN inG forma group, called the factor group G/N of G

over N.
http://find/http://goback/


5/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Factor Groups

Gis built up ofNand G/N in a certain way.

G/N

N {1}

{1}
http://find/


6/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Factor Groups

Gis built up ofNand G/N in a certain way.

{1}

G/N

N

{1}

{1}

N

G


7/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Factor Groups

We can repeat this process.

N1

N2

3N

N2

N1

N2

N1

3N

3N

simple

simple

simple

simple

{1}

G

G/

/

/
http://find/


8/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Groups of prime order

If|G|=pand pis prime then G is simple.

(1, 2, 3)={(), (1, 2, 3), (1, 3, 2)}issimple.
http://find/


9/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Alternating Groups

Sn contains a subgroup An consisting of all

permutations which can be expressed as anevennumber of 2-cycles.

An has half as many elements as Sn and is

simple. Sn/An contains two elements and isalso simple.
http://find/


10/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Classical Groups

Another important class of finite simplegroups arises from matrix groups.
http://find/


11/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Classical Groups

q finite field with qelements.V d-dimensional vector space over q.GL(d, q) group of invertibled d matrices

with entries in q.

GL(d, q) called GeneralLinear group.It is the group of all invertible lineartransformations from V to V.
http://find/


12/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

General Linear Group GL(2, 2)

GL(2, 2) =

1 00 1

,

1 10 1

,

1 01 1

,

0 11 0

, 0 11 1

, 1 11 0
http://find/


13/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

General Linear Group

How many elements does GL(d, q) have?

|GL(d, q)| equals the number of different

bases [v1, . . . , vd] for V. qd 1 choices for first vector

qd qchoices for second vector

qd q2 choices for third

...


14/52

Groups and Geometry

Alice Niemeyer

Groups


Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements




bases [v1, . . . , vd] for V.

|GL(d, q)| = (qd 1)(qd q) (qd qd1)

= qd(d1)/2d

i=1

(qi 1).
http://find/


15/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements




bases [v1, . . . , vd] for V.

gapSize( GL(4, 3 ) );> 24261120

gap36 * (3-1)*(32-1)*(33-1)*(34-1);> 24261120
http://find/


16/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Action on V

GL(d, q) maps vectors in Vto vectors in V:

gGL(d, q) maps vV to vg.

We say GL(d, q) acts on V.
http://find/


17/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Example action on V

Let =GF(2) and d= 2.Let

g=

0 11 1

and v= (1, 1). Then

vg= (1, 1)0 1

1 1 = (1, 0)
http://find/


18/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Special Linear Group

SL(

d,

q) is the subgroup of

GL(

d,

q)consisting of all elements with determinant 1.

|GL(d, q)|= (q 1)|SL(d, q)|.
http://find/


19/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Action on subspaces ofV

GL(d, q) maps 1-dimensional subspaces of

Vto 1-dimensional subspaces:x is mapped by g toxg.

P(V) ={v |v=0}

is the set of 1-dimensional subspaces.
http://find/


20/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Projective Spaces

A projective space consists of points, lines,planes,etc.

Points: 1-dimensional subspaces

Lines: 2-dimensional subspaces

Planes: the 3-dimensional subspaces

etc
http://find/


21/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Projective Spaces

The point v lies on the line x, y if

v x, y.

The line U is contained in the plane W ifUW.
http://find/


22/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Projective Spaces

The incidence structure is defined by inclusi-on.

G d G
http://find/


23/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

GF(2)4 generates PG(3, 2)

points

lines

planes

G d G t
http://find/


24/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

PG(d 1, q)

PG(d 1, q) is projective space generatedby the vector space Vof dimension d.

We say that the projective dimension ofPG(d 1, q) is d 1.

Groups and Geometry
http://find/


25/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Action on the Projective

Space

GL(d, q) acts on the projective space:

gGL(d, q) maps the r-dimensionalsubspace U=u1, . . . , ur to another

r-dimensional subspace, namelyUg=u1g, . . . , urg.

Groups and Geometry
http://find/


26/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Example

Let =GF(2) and d= 4.

g=

0 0 1 11 1 0 00 1 1 00 0 1 0

acts on the 2-dimensional subspaceU = (0, 1, 1, 0), (1, 0, 0, 0) by

Ug = (1, 0, 1, 0), (0, 0, 1, 1).

Groups and Geometry
http://find/


27/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Elements that do nothing

Elements g=aI fora

fix eachr-dimensional subspace. Hence they acttrivially on the projective space.

All other elements act non-trivially.

Groups and Geometry
http://find/


28/52

Groups and Geometry

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

The Centre ofGL(d, q)

The center ofGL(d, q) is the set of allmatrices that commute with all elements of

GL(d, q).

Z(GL(d, q)) ={aI|a },

so the centre consists of the elements thatact trivially on the projective space.

Groups and Geometry
http://find/


29/52

p y

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Projective Linear Groups

PGL(d, q) =GL(d, q)/Z(GL(d, q))

is the projective linear group.

PSL(d, q) =SL(d, q)/Z(SL(d, q)).

is the projective special linear group.

Groups and Geometry
http://find/


30/52

p y

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Simplicity of Projective

Special Linear Groups

For d2 and (d, q)= (2, 2), (2, 3)PSL(d, q) is simple.

Groups and Geometry
http://find/


31/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Collineations

A permutation gof the points is acollineationof a projective space, if it mapslines to lines.

Collineations map subspaces to subspaces.

Groups and Geometry
http://find/


32/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Automorphism Groups of

Projective Spaces

The automorphism groupofPG(d 1, q) isthe set of all collineations.

Note that PGL(d, q) is a subgroup of the full

automorphism group ofPG(d 1, q) ford3.

Groups and Geometry
http://find/


33/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Full automorphism group

The full automorphism group of

PG(d 1, q) for d3 is PL(d, q) and it isnot much larger than PGL(d, q).

Note PL(d, q) =PGL(d, q) when q is

prime.

Groups and Geometry
http://find/


34/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

How do we prove this?

Base and Stabiliser Chain argument.

Groups and Geometry
http://find/


35/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Groups and Projective

Spaces

Through the study of projective spaceswe learn about their automorphismgroups.

Through the study of the automorphismgroups we learn about the udnerlyinggeometries.

Groups and Geometry
http://goforward/http://find/http://goback/


36/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Other subgroups

GL(d, q) has other important subgroups.

These arise as subgroups which preservecertain types of inner products.Here we only look closely at one example,namely the symplectic group.

Groups and Geometry
http://find/


37/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Polar Spaces

We define a new incidence structure fromPG(d 1, q) by deleting various subspaces.

Groups and Geometry


38/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Symplectic Forms

f :V V is symplectic if for allu, v, wV and a

f(u + v, w) =f(u, w) +f(v, w),

f(u, v+ w) =f(u, v) +f(v, w),

f(au, v) =f(u, av) =af(u, v),

bilinear form

Groups and Geometry


39/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Symplectic Forms

And also

f(u, v) =f(v, u),

f(x, y) = 0 for all xV implies y= 0,(non-degenerate)

f(x, x) = 0 for all xV.We write uv if f(u, v) = 0 and say u isorthogonalto v.

Groups and Geometry
http://find/


40/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Example of Symplectic Form

Let

A=

0 0 0 10 0 1 0

0 1 0 01 0 0 0

Define a form f on V =GF(2)4 by

f(u, v) =uTAv.

Then f is a symplectic form.

Groups and Geometry


41/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Symplectic Groups

gGL(d, q) preservessymplectic form f if

f(ug, vg) =f(u, v) for all u, vV.Define

Sp(d, q) ={gGL(d, q)|g preserves f}.

Sp(d, q) is the symplectic group.

Groups and Geometry
http://find/


42/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Symplectic Polar Spaces

Let f :V V be a symplecticform.The symplectic polar space SP(d 1, q)consists of those subspaces S ofPG(d 1, q) for which

f(u, v) = 0 for all u, vS.

Groups and Geometry
http://find/


43/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

SP(4, 2) where V=GF(2)4

Groups and Geometry


44/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

SP(4, 2) where V=GF(2)4
http://find/


45/52

Groups and Geometry


46/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Rank of a polar space

The rankof a polar space is the number ofdifferent types of objects (points, lines,

planes) we have.

By a theorem of Witt the rank of a polarspace is at most d/2.

rank ofSP(4, 2) is 2.

Groups and Geometry

Ali Ni
http://find/


47/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements


Generalised Quadrangles are polar spaces ofrank 2.

Groups and Geometry

Ali Ni


48/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Axioms for Generalised

QuadranglesA Generalised Quadrangleof order (s, t) isan incidence structure S= (P, L) consisting

of a point set Pand a line set Lsuch that Each point lies on t+ 1 lines;

Two distinct points are on at most one

line; Each line has s+ 1 points;

Two distinct lines have at most one

point in common; Groups and Geometry

Alice Niemeyer
http://find/


49/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

The GQ-Axiom

For a lineand a point Pnot on there is auniqueline m such that

P is on m;

m intersects in exactly one point.

Groups and Geometry

Alice Niemeyer
http://find/


50/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

The GQ-Axiom

Groups and Geometry

Alice Niemeyer
http://find/


51/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

The GQ-Axiom

Groups and Geometry

Alice Niemeyer
http://find/


52/52

Alice Niemeyer

Groups

Factor Groups

Classical Groups

Projective Spaces

Polar Spaces

Symplectic Groups


Acknowledgements

Acknowledgements

Many thanks to Dr John Bamberg and DrMaska Law for their help in preparing theseslides and the accompanying workshop.
http://find/

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Niemeyer Groups and Geometry Slides 2004

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