I. The B u r n s i d e
In this
section
some of the
Burnside
rin~ of finite
let G denote
subsequent
by g e o m e t r i c
always
suitable
tation
theory.
a finite
investigations
ring of a finite
Lie groups
G-sets.
group.
we give
In order
which
for the a p p l i c a t i o n s
in this
to m o t i v a t e
an i n t r o d u c t i o n
Later we g e n e r a l i z e
methods
The m a t e r i a l
group.
this to compact
in case of a finite
of the B u r n s i d e
section
is m a i n l y
to the
group
ring
are not
in represen-
due to A n d r e a s
Dress.
1.1. F i n i t e
A finite
this
G-sets.
G-set S is a finite
set. A finite
are t r a n s i t i v e
G/H =
G-sets
{gHlg E G~
a commutative
arises
diagonal
the orbit
space
the G - i s o m o r p h i s m
to X = G/K.
through
1.2.
A+(G)
with
G/H x X
) GXHX
left H-action.
with addition
by c a r t e s i a n
pro-
These
orbits
can be i d e n t i f i e d
This
correspondence
with
can
the H - o r b i t s
of X corres-
X is a G-space
then we have
: (g,x)~------>(g,g-lx).
the double
of finite
of the m u l t i p l i c a t i o n
g E G, w h i c h
If m o r e o v e r
if and only
classes
x G/K into orbits.
If X is an H-space
The orbits
G-sets
identity
induced
of G on
We apply
coset HgK c o r r e s p o n d s
this
to the orbit
(1,g).
The B u r n s i d e
The G r o t h e n d i e c k
A(G)
HgK,
the
of GXHX.
Explicitely,
set of G - i s o m o r p h i s m
of G/H
cosets
as follows:
to h o m o g e n e o u s
The n o n - t r i v i a l i t y
of G/K under
to the G - o r b i t s
union of its orbits.
and m u l t i p l i c a t i o n
action.
to the d o u b l e
be d e s c r i b e d
pond
union
a left action
G/H and G/K are i s o m o r p h i c
semi-ring
from the d e c o m p o s i t i o n
correspond
with
and are G - i s o m o r p h i c
to K in G. The
induced by d i s j o i n t
duct w i t h
is the d i s j o i n t
. The G-sets
if H is c o n j u g a t e
G-sets b e c o m e s
G-set
set t o g e t h e r
rin@ A(G).
ring
constructed
and will be called
from the s e m i - r i n g
the B u r n s i d e
A+(G)
is d e n o t e d
rin~ of G. If S is a finite
G-set
let
IS] or S be its image in A(G).
group on isomorphism classes
additive
C(G)
Z-basis
Example
A(G)
with unit
is the free abelian
G-sets.
where
Equivalently,
an
(H) runs through the set
of G. The m u l t i p l i c a t i o n
of G/H x G/K into orbits.
The ring A(G)
comes
is
[G/G] .
1.2.1.
Let G be a b e l i a ~ Then,
at
~/H]
classes of subgroups
from the decomposition
commutative
of transitive
is given by the
of conjugacy
Additively,
since generally
-I
(gl H, g2 K) is giHgl ~
abelian case. Therefore
tained by counting
[G/HI 2 =
groups are H ~ K in the
[G/H]" [G/K] = a [ G / H ~ K] where a ~ Z is ob-
the number of elements
IG/H I [G/HI
for abelian G the
the isotropy group of G/H x G/K
-1
g2Kg2 , all isotropy
, where
[G/~
on both sides.
ISI is the cardinality
In particular
of S. We see that
are almost idempotent.
If H < G and S,T are finite G-sets then we have for the cardinality
of the H-fixed point sets
IsHI ITHI. Hence S ,
%IsHI extends
~H
Conjugate
for each
subgroups
IS H + T H I = IsHI + ITHiand
: A(G)
I(S x T ) H I =
to a ring h o m o m o r p h i s m
.............. --)
Z
give the same h o m o m o r p h i s m
so that we have one
(H) E C(G). We let
= ( ~ H ) : A(G)
be the product of the
Proposition
--'>
~H"
1.2.2.
is an in~ective
rin~ h o m o m o r p h i s m .
~(H)
& C(G)
Z
~H
Proof.
By definition ~ is a ring homomorphism.
Suppose x # o is in the kernel
of ? . We write x in terms of the basis x = ~ aH[G/H].
ordering on the
conjugate
[G/~
, namely
[G/HI
~
to K. Let [G/HI be maximal
a H ~ o. Then G/K H # ~ implies
[G/K]
We have a partial
if and only if H is sub-
among the basis elements with
~/H]
~
[G/K]
. Hence o =
~H x =
= a H [G/H H] = a H INH/H I # O, a contradiction.
Since ~ is an injection of a subgroup of maximal
is a finite group.
We want to compute
its order.
rank the cokernel
We consider
the
diagram of injective ring homomorphisms
)T[z
A(G)
!
$
A(G) ~ Q
~Q
where
the lower
~Q
is the rational extension of the upper
Recall that WH = NH/H acts
morphisms:
freely on G/H as the group of G-auto-
The action is given by WH X G/H - > G/H
: (wH,gH) ~--> gw
Hence it acts freely on any f i x e d point set G/H K. In particular
is divisible by
Proposition
The elements
IWHI
. Therefore
-~Q( [G/HI ~
--
IWHJ -1)
-1
H.
IG/HKI
is contained
~Z.
1.2.3.
~e(
order of c o k e r n e l ~
[G/HI
~
IWH~ -I) =: x H form a Z-basi____~so f
i__ss ~(H) E C(G)
IWHI
Z. The
Proof.
The
first
assertion
as row vectors.
with
one's
Remark
implies
Then
the x H form
on the diagonal.
as follows.
~
Hence
ordered)
elements
in
• Z
a triangular
matrix
they must be a basis.
may be d i s c o v e r e d
An e l e m e n t
x ~ A(G)
x has no zero component.
ring of A(G)
x G A(G) ~
integral
(i.e.
over
integral
of A(G)
notion
over
z hence
[3~]
Ch.
seen
between
in 1.2.
~
~z
of
elements
in its total q u o t i e n t
see Lang
~0~j,
If
~A(G),
are
the i n c l u s ~ n
ring.
Chapter
are
over
which
with
if
quotient
~QX
is integral
may be i d e n t i f i e d
order p then
~sGI
ISI ~
p. Hence
be in the image of
only condition,
~
image?
IX;
(For the
Bourbaki
is a su b g r o u p
mod p b e c a u s e
congruence
for G = Z/pZ.
a linear action
numbers.
If G = Z/pZ
. The reader
G-set
by the elements
is a f u n c t i o n
~ A(G)
this
Let S be a finite
the p e r m u t a t i o n
fixed p o i n t
that
its
spanned
closure
invertible).
then the c o m p o n e n t s
by i d e m p o t e n t
Hence
if and only
Q is the total
made
Conversely
ring e x t e n s i o n
How can we d e s c r i b e
cardinality
A(G) ~
of A(G)
5.)
Congruences
We have
integers.
into the i n t e g r a l
,
Therefore
over A(G)
any subring.
structure
is a n o n - z e r o d i v i s o r
~ Z is g e n e r a t e d
of integral
from the ring
all n o n - z e r o - d i v i s o r s
Q is integral
e. g. b e c a u s e
duces
(suitably
We view
1.2.4.
The h o m o m o r p h i s m
1.3.
the second one.
and
the orbits
can easily
We g e n e r a l i z e
representation
The
check
for e l e m e n t s
that this
vector
on the basis
resulting
is the
G-module
the same
space
S of V(S)
V(S)
in-
is called
to S. The c h a r a c t e r
on G; it will be d e n o t e d w i t h
to
such congruences.
be the complex
associated
in ~ Z .
of S ~ S G have
a condition
of S. The G - a c t i o n
on V(S).
rank
is the cyclic g r o u p of p r i m e
gives
let V(S)
of m a x i m a l
symbol.
of V(S)
The
orthogonality
relations
for characters
complex G-module V the number
say in p a r t i c u l a r
IGI -I ~
g E G V(g)
that for any
is the dimension
of
V G. Hence
(1.3.1)
~ g E G V(S) (g) =-
O
mod
~G I .
Now note that
V(S) (g) = Trace(ig
(look at the matrix of i
: V(S)
.~V(S)
: v ~----> gv)
= Isgl
g with respect to the basis S). Therefore
1.3.1
can be rewritten
(1.3.2)
I
gEG
~<g>
(x) ---- O
for any x 6 A(G), where < g > denotes
mod ~GI
the cyclic group generated by g.
If H is a cyclic subgroup of G the number of elements
conjugate
g with { g >
to H is
IH L JG/NH I
where H ~ i s
groups
the set of generators
conjugate
(1.3.3)
~"
to H. Therefore
(H) cyclic
where now the summation
of H and IG/NH I is the number of
(1.3.2)
IH~(-i IG/NH i ~ H
can be rewritten
(X) ~ 0
is taken over conjugacy
mod
IGI
classes of cyclic sub-
groups of G.
We now apply the same argument
and obtain
to V(S H) considered
as N H / H - m o d u l e
7_
where we
(x>
sum over N H - c o n j u g a c y
and K/H is cyclic.
This may
(I .3.4)
(K) n(H,K)
where
Z
the n(H,K)
~ K
(x) ~
integers
INH/HI
mod
o
K such that H is normal
also be w r i t t e n
are certain
over the G - c o n j u g a c y
classes
=
in K
in the form
0
mod INH/H I
w i t h n(H,H)
= I and the sum is
classes(K) such that H is normal
in K and K/H is
cyclic.
For the next P r o p o s i t i o n
C (S) - - +
we view elements
~Z
as functions
Z.
Proposition
1.3.5.
The c o n @ r u e n c e s
1.3.4
are a complete
i. e. x ~ ~ Z is c o n t a i n e d
set of c o n @ r u e n c e s
in the image
(K) n(H,K)
for all
of
x(K)~
o_~f ~
O
mod
for image
if and only
~
,
i_~f
~NH/H I
(H) & C(G).
Proof.
We have
already
congruences.
seen that
The c o n g r u e n c e s
given by a t r i a n g u l a r
describe
a subgroup
fore A = im
the elements
1.3.6
matrix with
A of index
~
in the image of
are i n d e p e n d e n t
one's
~
satisfy
because
on the diagonal.
INH/H I . By P r o p o s i t i o n
these
they are
Hence
1.2.3
they
there-
Remark
1.3.7.
A slightly
V(S H)
different
set of c o n g r u e n c e s
as N p H / H - m o d u l e
where
yields
a set of p - p r i m a r y
1.3.4.
These
considered;
p-primary
1.4.
Idempotent
Idempotent
use
elements
of
~
of NH/H.
may be used
localizations
the B u r n s i d e
ring
This
instead
of A(G)
localized
of
are
at p, o n l y
are valid.
in
~
such
Z are
the
functions
H of G is c a l l e d
Each
is s o l v a b l e .
when
p-group
considers
functions
come
with
f r o m A(G).
values
0 and
We c o n s i d e r
I. We
A(G)
as
Z via
A subgroup
subgroup.
which
if one
elements.
1.3 to see w h e n
subring
are u s e f u l
for A(G) (p),
congruences
is a S y l o w
congruences
congruences
e. g.
NpH/H
is o b t a i n e d
H < G has
One
has
if H = H s. L e t P(G)
perfect
a smallest
if it is e q u a l
normal
subgroup
(Hs) s = H s. A s u b g r o u p
be the
subset
of C(G)
to its
H s such
H is p e r f e c t
represented
commutator
that
H/H s
if and o n l y
by p e r f e c t
sub-
groups.
Proposition
1.4.1.
An idempotent
(H) E
C(G)
e
~
~ Z is c o n t a i n e d
the e q u a l i t y
e(H)
in A(G)
if and o n l y
if for
all
= e(H s) holds.
Proof.
Suppose
e ~ A(G).
K s = K n 4 K n-1 4
p(i).
T h e n by
m o d p(i).
and
e satisfies
... 4 K ° = K such
1.3.6
Since
therefore
Then
applied
the v a l u e s
K. T h e n we m u s t
have
e satisfies
congruences
the
that
group
Conversely
= e(K)
Given
K i / K i+I
of e are 0 or
e(K s) = e(K).
e(H)
to the
1.3.6.
< G. C h o o s e
of
is c y c l i c p r i m e
K i+I w e h a v e
I we m u s t
assume
for all H ~
1.3.6.
K
e(K i) ~
have
that
e(K s)
K with
K/H
order
e(K i+I)
e(K i) = e(K i+I)
= e(K)
cyclic
for all
so that
Corollary
The
1.4.2.
set of i n d e c o m p o s a b l e
t__oo P(G).
only
I_nn p a r t i c u l a r
idempotents
Remark
Z be
G is s o l v a b l e
corresponds
if and o n l y
bi~ectively
i_~f 0 and
I are the
i__nnA(G).
a set of p r i m e
A(G)
at P,
show
as in the
i. e.
are
the
primes
in P.
not
Let A(G)p
in P are m a d e
of P r o p o s i t i o n
functions
subgroup
numbers.
the p r i m e s
proof
normal
1.5.
of A(G)
1.4.3.
Let P C
A(G)p
idempotents
e with
of H s u c h ~ a t
e(H)
1.4.1
localization
invertible.
that
= e(Hp)
H/Hp
be the
the
where
is s o l v a b l e
Then
one
idempotents
Hp
is the
of o r d e r
of
can
of
smallest
involving
only
Units.
If A is a c o m m u t a t i v e
ring we
let A ~ be the m u l t i p l i c a t i v e
group
of its
units.
Let
e ~ A be an i d e m p o t e n t .
can h a p p e n
that
for a unit
Then
e is an i d e m p o t e n t ,
case
of the B u r n s i d e
general
then
coker ~
(I-u)/2
by
in A(G)
is o d d
E A(G).
1.4.2,
we
Proposition
and h e n c e
= u is a unit.
(l-u)
2
(I-u)/2
= 2(I-u)
is c o n t a i n e d
in a m o m e n t .
1-u
E A(G)
a non-solvable
group
and
= e is c o n t a i n e d
for any u n i t
in
But
~
u.
Z but not
if G has
(I-u)/2
has
Conversely
it
in A.
In
in
odd o r d e r
E ~ Z implies
non-trivial
idempotents,
obtain
1.5.1.
t h e n A(G) ~
L e t H be
see
I-2e
element
(I-u)/2
shall
Since
If G is n o n - s o l v a b l e
order
u the
because
ring
as we
Then
=
then
{ ~
a subgroup
I}
A(G)
{ +
I ~
. If G is s o l v a b l e
of odd
.
of i n d e x
2 in G. T h e n
H 4 G,
[G/HI
2 = 2 [GIH]
and therefore u(H)
in A(G).
:= I - [G/HI
~
A(G) W. Note that
(1-u(H))/2 is not
The subgroups of index 2 are p r e c i s e l y the kernels of non-
trivial h o m o m o r p h i s m s G---~Z/2Z.
j : Hom(G,Z/2Z)
) A(G) ~
Hence we obtain an injective map
given by j(f) = 1-G/ker(f) . The image of j
is in general not a subgroup.
P r o b l e m 1.5.2.
Determine the structure of A(G) ~ in terms of the structure of G.
(Of
course one knows by the famous theorem of Feit - T h o m p s o n that groups
of odd order are solvable.
relevant.
T h e r e f o r e the 2-primary structure of G is
In p a r t i c u l a r A(G) ~ for 2-groups w o u l d be interesting.
(See
also the next remark.)
Remark
1.5.3.
We shall prove later by g e o m e t r i c methods that for a real r e p r e s e n t a t i o n
V
the function
(H) I
)(-I) dim vH is c o n t a i n e d in A(G).
is then a unit in A(G).
not of this form
This function
It w o u l d be i n t e r e s t i n g to see units w h i c h are
(2-groups?).
1.6. Prime ideals.
Since
~ Z is integral over A(G)
Seidenberg
A(G)
(see A t i y a h - M a c
comes from
Donald
:=
{ x ~ A(G)
for a subgroup H of G and a prime
[~]
[11]
, p. 62) every prime ideal of
~ Z hence has the form
q(H,p)
of Dress
by the "going-up theorem" of Cohen-
I
ideal
~H
(x) -= 0
rood p }
(p) of Z. The e l e m e n t a r y proof
for this fact shall be given later
(section 5) in the
slightly more general context of compact Lie groups. The p r i m e
q(H,o)
are minimal;
field Z/pZ.
ideals
the ideals q(H,p), p # O, are maximal w i t h residue
If q(H,p)
= q(K,q)
then p = q and
10
(i)
(H)
= (K)
if
p = 0
(ii)
(H)
P
=
if
p # O.
(K)
P
Here Hp is the smallest normal subgroup of H such that H/Hp is a pgroup.
mal
If q is a prime ideal of A(G)
(H) such that
where
[G/~
~
then there exists a unique mini-
q. M o r e o v e r for this H one has q = q(H,p)
p is the c h a r a c t e r i s t i c of the ring A(G)/q.
the m a x i m a l
(H) for w h i c h q = q(H,p).
Finally this
(H) is
All this is proved in Dress
[~
and w i l l later be p r o v e d for compact Lie groups.
1.7. An example:
The a l t e r n a t i n g group A 5.
The d i a g r a m of c o n j u g a c y classes of subgroups of A 5 is
A5
Ds
I
Here D n is~ the d i h e d r a l group of order 2n. The groups A5, A4, D5, D 3
are their own n o r m a l i z e r s while N(Z/n)
set of functions
(i)
(ii)
(iii)
(iv)
z(H)
= D n and N(D2)
=A 4. A(A 5) is the
z : C(G)---) Z satisfying
a r b i t r a r y for H = A 5 , A 4 , D 5 , D 3.
z(Z/n) ~
z(D 2) m
z(D n) mod 2
for
n = 3,5.
z(A 4) mod 3.
z(1) + 2Or(Z/3)
+ 15z(Z/2)
+ 24z(Z/5) ~ 0 mod 60.
The ring A(A 5) contains the following units:
11
I
Z/2
Z/3
Z/5
D2
D3
D5
A4
A5
a
a
a
a
b
c
d
b
e
Here a , b , c , d , e
function
6
{I,-I~
u : C(G) ---9 Z at the element
congruences
(i) -
(iv)
at A 5, A 4, D 3. From
mod
3, mod
subgroups
idemp o t e n t s
1.8.
role
indicated
there
the value
in the first
are no conditions
u(D 2) = u(A4).
of the
line.
The
for a unit u
Considering
(iv)
= u(Z/2)
= u(Z/3)
= u(Z/5).
I and A 5 are perfect.
where
~A5(e)
Therefore
= I,
A ( A 5) contains
the
~H(e)
= O for H # A 5.
[~]
where
Comments.
ring was
s p e c t r u m was
introduced
determined.
in the a x i o m a t i c
in p a r t i c u l a r
The B u r n s i d e
of Dress,
in the general
ring,
simplicial
ring.
simplicial
theory
(Green
of i n d u c t i o n
is u n i v e r s a l
also the prime
an i m p o r t a n t
[~S]
theorems
, Dress
[~0 3 )
(Dress [gO]).
among the M a c k a y
functors
references.
in these
lectures
simplicial
G-sets.
G-complexes
Characteristic":
theory
ring plays
the t o p o l o g i c a l
At this p o i n t we only m e n t i o n
complex with
from finite
The B u r n s i d e
as a functor,
demonstrate
of the B u r n s i d e
by Dress
representation
see the cited
We shall
built
line gives
5 we o b t a i n
O,1,e,l-e
The B u r n s i d e
ideal
show that
(iii) we o b t a i n
4, and m o d
u(1)
The
and the second
G-action
So one expects
sum
~
a finite
is a c o m b i n a t o r i a l
some b a s i c
to lie in the B u r n s i d e
the a l t e r n a t i n g
that
significance
ring,
(-1)iSi
invariants
e.g.
the
object
of
"Euler-
of the G-sets
S i of
i-simplices.
The B u r n s i d e
properties
ring
codifies
of the lattice
in a c o n v e n i e n t
of subgroups
frame-work
of a g i v e n group.
some basic
Given
G, the
12
G-transformation
Burns i d e
ring.
w h e n we have
homotopy
groups
This
influence
shown that
of sphere
lar stable
are g o v e r n e d
internal
of the B u r n s i d e
the ring
in d i m e n s i o n
equivariant
by the
homotopy
ring
is isomorphic
zero
is more
) so that
are m o d u l e s
of the
transparent
to e q u i v a r i a n t
(Segal [ I ~ 5 ]
groups
relations
stable
in p a r t i c u -
over the B u r n s i d e
ring.
The d e s c r i p t i o n
cardinalities
by Dress.
where
of the B u r n s i d e
of fixed p o i n t
These
congruences
also various
sets
ring using
is b a s e d
on an oral
are g e n e r a l i z e d
geometrical
congruences
among
communication
in tom D i e c k - P e t r i e
applications
[&$]
are given.
1.9. Exercises.
I. Let G and H be finite
Show
groups w h o s e
orders
are r e l a t i v e l y
prime.
that
A(G ~ H) ~ A(G) ~
2. For
i ~ O mod p
M(i~
Show that M(i)
modules
=
A(H)
let
~(a,b~ I a i ~
is a p r o j e c t i v e
b
mod p ~
module
c
Z ~ Z.
over A(Z/pZ).
Classify
projective
over A(Z/pZ).
3. Show that G is p e r f e c t
if and only
if A(G)
contains
the
idempotent
e such that
He = O
4. Let G be a p - g r o u p
for
H # G,
of o r d e r pn
~G e = I.
(p a prime).
Let m C A(G)
be the
ideal
m =
Show that m n + 1 C
topologies
{x
p A(G).
coincide.)
I ~{i ~ X = O
mod p ~
(In particular:
.
The p-adic
and the m - a d i c
13
5. Let G be a 2 - g r o u p
t h a n the n u m b e r
and let I A ( G ) ~ [ = 2 n. S h o w that n is not g r e a t e r
of c o n j u g a c y
classes
(H) s u c h t h a t
INH/H]
= 2.