Niemeyer Groups and Geometry Slides 2004
Transcript of Niemeyer Groups and Geometry Slides 2004
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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
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Groups and Geometry
Alice Niemeyer
Groups
Factor GroupsClassical Groups
Projective Spaces
Polar Spaces
Symplectic Groups
Generalised Quadrangles
Acknowledgements
Simple Groups
In Chemistry compounds are made up ofelements.
Is there an analouge for finite groups?
YES! Simple groups are the building blocks.
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Symplectic Groups
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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.
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Factor Groups
The cosets of a normal subgroupN inG forma group, called the factor group G/N of G
over N.
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Factor Groups
Gis built up ofNand G/N in a certain way.
G/N
N {1}
{1}
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Acknowledgements
Factor Groups
Gis built up ofNand G/N in a certain way.
{1}
G/N
N
{1}
{1}
N
G
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Generalised Quadrangles
Acknowledgements
Factor Groups
We can repeat this process.
N1
N2
3N
N2
N1
N2
N1
3N
3N
simple
simple
simple
simple
{1}
G
G/
/
/
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Symplectic Groups
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Acknowledgements
Groups of prime order
If|G|=pand pis prime then G is simple.
(1, 2, 3)={(), (1, 2, 3), (1, 3, 2)}issimple.
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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.
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Acknowledgements
Classical Groups
Another important class of finite simplegroups arises from matrix groups.
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Generalised Quadrangles
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.
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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
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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
...
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Alice Niemeyer
Groups
Factor GroupsClassical Groups
Projective Spaces
Polar Spaces
Symplectic Groups
Generalised Quadrangles
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.
|GL(d, q)| = (qd 1)(qd q) (qd qd1)
= qd(d1)/2d
i=1
(qi 1).
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Symplectic Groups
Generalised Quadrangles
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.
gapSize( GL(4, 3 ) );> 24261120
gap36 * (3-1)*(32-1)*(33-1)*(34-1);> 24261120
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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.
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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)
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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)|.
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Generalised Quadrangles
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.
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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
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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.
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Projective Spaces
The incidence structure is defined by inclusi-on.
G d G
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Acknowledgements
GF(2)4 generates PG(3, 2)
points
lines
planes
G d G t
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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.
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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.
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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).
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Generalised Quadrangles
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.
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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.
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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.
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Acknowledgements
Simplicity of Projective
Special Linear Groups
For d2 and (d, q)= (2, 2), (2, 3)PSL(d, q) is simple.
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Acknowledgements
Collineations
A permutation gof the points is acollineationof a projective space, if it mapslines to lines.
Collineations map subspaces to subspaces.
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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.
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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.
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How do we prove this?
Base and Stabiliser Chain argument.
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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.
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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.
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Acknowledgements
Polar Spaces
We define a new incidence structure fromPG(d 1, q) by deleting various subspaces.
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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
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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.
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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.
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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.
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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.
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Acknowledgements
SP(4, 2) where V=GF(2)4
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SP(4, 2) where V=GF(2)4
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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.
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Acknowledgements
Generalised Quadrangles
Generalised Quadrangles are polar spaces ofrank 2.
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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
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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.
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The GQ-Axiom
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The GQ-Axiom
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Acknowledgements
Acknowledgements
Many thanks to Dr John Bamberg and DrMaska Law for their help in preparing theseslides and the accompanying workshop.
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