Publicacions
Ma emá iques,
Vol
34
(1990),
323-333
.
ON
THE
CONNECTION
BETWEEN
THE
TOPOLOGICAL
GENUS
OF
CERTAIN
POLYHEDRA
AND
THE
ALGEBRAIC
GENUS
OF
THEIRHILTON-HOPF
QUADRATIC
FORMS
Abs ac
IMRE
BOKOR
The
Hil on-Hop
quad a ic
o m
is
de ined
o
spaces
o
he
homo opy
ype
o
a
CW
complex
wi h
one
cell
each
in
dimensions
0
and
4n,
K
cells
in
dimension
2n
and
no
o he
cells
.
I
wo
such
spaces a e
o
he
same
opological
genus,
hen
hei
Hil on-Hop
quad a ic
o ms
a e
o
he
same
weak
algeb aic
genus
.
Fo
la ge
classes
o
spaces,
such
as
simply
connec ed
di e en iable
4-mani olds,
he
con e se
is
also
ue,
as
long as
he
suspensions
o
he
spaces a e
also
o
he
same
opological
genus
.
This
no e
allays
he
conjec u e
ha
he
con e se
is
ue in
gene al
by
o e ing
wo
echniques
o
gene a ing
in ini e
amilies
o
coun e examples
.
I
was
shown
in
[1]
ha
o
C,',
he
class
o
hose
spaces
which
a e o
he
same
homo opy
ype
as a
CW-complex
wi h
p ecisely
one
cell
in
each
o
he
dimensions
0,
2n
and
4n,
he e
is
a
Fac o isa ion
Theo em
wi h
espec
o
he
one-poin
union
(o
V-p oduc )
o
spaces
.
The
heo em
s a es
ha
as
long
as
he a aching
map
o
he
4n-cell
ep esen s
a
homo opy
class
o
in ini e
o de
in
7 4n-1(S2n),
he
homo opy
ype
o
a
ini e
wedge
o
such
spaces
de e mines
he
homo opy
ype
o
each
o he
spaces
in
he
wedge
.
This
s ands
in
con as
o
he
si ua ion
whe e
he a aching
map
is
o
ini e
o de , o
hen
non-cancella ion
phenomena
occu
wi h
spaces
o
he
same
"genus",
as
shown
in
[31,
[4], [5]
and
[6]
.
The
p oo
o
he
Fac o isa ion
Theo em
applied
echniques
which
sugges
a
gene alisa ion
o
he
cha ac e isa ion
o he
genus
o
a
space
in
C,'
o
Cñ
,
he
class
o
spaces
o
he
homo opy
ype
o
a
CW-complex
wi h
one
cell
in
dimensions
0
and 4n and a
ini e
numbe
K
o
cells
in
dimension
2n
.
This
la ge
class o
spaces
includes
all
(2n
-
1)-connec ed
di e en iable
4n-mani o1ds
.
Up
o
homo opy,
he
spaces
in
Ch
a e
mapping
cones
C
o
con inuous
maps
K
;
S4n-1

V
s2n
k=1
324

1
.
BOKOR
K
and
hence
a e
classi ied
by
7 4n-1(
V
S
2n
)
.
k=1
K
We
ix
K
and
abb e ia e
V
S2n
o
V
S2n
.
k=1
Fo
maps
,
g
:
Son-1
-->
V
S2n
he
se
o
homo opy
classes
o
con inuous
maps
be ween
hei
mapping
ones
C
and
C
gis
in bijec ion
wi h
he
se
o
homo opy
classes
o
homo opy
commu a i e
diag ams
whe e
and
S4n-1

i
S2n
9
w
The
se
o
homo opy
classes
o
maps
V
S2n
--,
V
S
2n
admi s
a
ing
s uc u e
isomo phic
o
M(K
;
Z),
he ing o
K
x
K
in eg al
ma ices
.
The
unc ion
assigning
o each
sel -map
cp
o
S
2
n
i s
deg ee
induces
one
such
isomo phism,
by
mapping
cp
o he
ma ix
A(cp)
whose
(i,
j)- h
eoe lcien
is
he
deg ee
o
he
composi e
map
qi
o
(p
oi dj
:
SU-)
V
S2n
-a
V
S2n
-~
S2n,
2nj
:
S2n
-4
V
S
2
n
is
he
.
j- h
canonical
inclusion in
he
co-p oduc
and
qi
S
2n
->
S2n
collapses
each
summand
o he
base
poin
excep he
i- h
one,
on
which
i
ac s
as
he
iden i y
map
.
I
was
shown
in
[1]
ha
each
homo opy
class
[ ]
E
7 4n-1(V
S
2n
)
can
be
ep esen ed
by a
pai
consis ing
o
he
Hil on
-
Hop
quad a ie
o m
o
and
he
suspension
o
.
The
Hil on-Hop
quad a ic
o m
can
i sel
be
ep esen ed
as
a
symme ic
in eg al
K
x
K
ma ix,
HM,
whose
en ies
a e p ecisely
he
Hil on-
Hop
in a ian s
o
,
and
he
suspension
by
E
,
a
"column
o sion
ec o "
each
o
whose
componen s
is
an
elemen
7 4n(S2n+1
)
.
A
sui able
choice
o
gene a o s
o
7 4n-1(V
S2n)
p o ides
a
simple
way
o
compu ing
cp o
om
co
and
by
means
o a
ma ix
calculus
:
H(
,
P o
)
=
A(~P)H(
.
)(A(~P))
E(cp o
. )
=
AME( )
)
.
and
TOPOLOGICAL
AND
ALGEBRAIC
GENUS
OF
POLYHEDRA

325
Thus
H( )
is
a
quad a ic
o m
which
is
an
in a ian
o
o ien ed
homo opy
ype
and
he
assignmen
o
each
E
7 4n-1
(
S
2
n)
o
i s
Hil on-Hop
quad a ic
o m
-
ha
is,
he
quad a ic
o m
wi h ma ix
H( )
in
he
app op ia e
basis
-
is
na u al
in
sel -maps
o
V
SU
.
(Re an
ha
he
Hil on-Hop
quad a ic
o m
can
also
be
hough
o as
he
in e sec ion
o m
in
he
in eg al
cohomology
o
C
.)
The
genus
o
a
nilpo en
space
- mo e
p ecisely,
he
genus
o
he
homo opy
ype
o
a
CW-complex
o
ini e
ype
-
is
de ined
o
be
he
se
o
homo opy
ypes
o
hose nilpo en
spaces,
each
o
whose
p-localisa ions
is
homo opy
equi alen
o
he
p-localisa ion o
he
o iginal
space
.
Since
we
a e
only
conce ned
wi h
mapping
cones
C
o
maps
:
Son-1
_>
V
S2n,
i
ollows
om
Lemma
2
.2
o
[1]
ha
he
wo
spaces
C
and
C
9
a e o
he
same
genus
i
and
only
i
o
each
p ime
p
he e
is
a
homo opy
commu a i e
diag am
s4n-1
i
2n
-i
s
2n
N
s4n-1

---------
4
VS2n
wi h
bo h
he
deg ee
o
0
and
he
de e minan
o
he
ma ix
p
.
In
e ms
o
he
ma ix
no a ion
de eloped,
his
diag am
equa ions
deg(O)H(g)
=A(So)H( )(A(w))`
deg(O)ag
=
A(~p)u ,
o
cp
cop ime
o
is
equi alen
o
wi h
deg(0)
and
de A(cp)
cop ime
o
p,
whe e
de
A
is
he
de e minan
o
he
ma ix
A
.
Since
he
assignmen
o
his
quad a ic
o m
is
na u al,
each
o
i s
in a ian s
is,
in
ac ,
an
in a ian
o
he
mapping
cone
o
.
One
such
in a ian
is
he
genus
o
a
quad a ic o m
.
An
immedia e
ques ion
is
:
Wha
is
he
connec ion
be ween
¡he
opological
genus
o
he
mapping
cone
o
and
he algeb aic
genus
o
á s
Hil on-Hop
quad a ic
o m
H( )
?
The
p esen
pape
is
de o ed
o
a
s udy
o his
ques ion,
using
he
no a ion
o
[1j
.
The
no ion
o
genus
in
he heo y
o
in eg al
quad a ic
o ms
has
a
simila
de ini ion,
namely
wo
in eg al
quad a ic
o as
1
and
V
o
dimension
K
a e
o
he
same
genus
p ecisely
when
o
e e y
p ime
p
hey
a e
equi alen
o e
Z
p
,
he
ing
o
p-adic
in ege s
.
We
ecall
he
de ini ion o
equi alence
o
wo
quad a ic
o ms
o e
a
ing in
i s
mo e
gene al
o m
.
De ini ion
.
Le
A
be
a
commu a i e
ing o cha ac e is ic
0,
so
ha
Z
is
a
sub ing
o
A
.
A
A-quad a ic
o m
o
dimension
K
is
a
unc ion
~D
:
M
-->
A
om
he ee
A-module
M
o
ank
K
o
A
sa is ying
<P(Ax)
=
A
2
$(x)
o
e e y
326

I
.
.
BoKOR
A
E
A
and x E
M
.
The
wo
A-quad a ic
o ms
1¿
:
M
-+
A and
V
:
M'

+
A
o
dimension
K
a e said o
be
A-equi alen
o
equi alen
o e
A
i
he e
is
an
isomo phism
:
M
-->
M'
such
ha
41
=
V
o
.
Obse e
ha
any
in eg al
quad a ic
o m
may
be
conside ed
o
be a
A-
quad a ic
o m
.
Hence
i
makes
sense
o
speak
o
A-equi alen e
o
in eg al
quad a ic
o ms
.
The
de ini ion
o
he
genus
o
an
in eg al
quad a ic
o m
conce ns
he
cases
A=Z
p
,
he
ing
o
p-adic
in ege s,
p
a
p ime
numbe
.
We
ansla e
his
de ini ion
in o
he
language
o
ma ices,
since
we
shall
be compu ing
wi h
ma ices
.
Two
in eg al
quad a ic
o ms
ep esen ed
by
symme ic
in eg al
ma ices
H
and H'
a e o
he
same
genus
i
and
only
i
o
each
p ime
p
he e
is
an
in e ible
p-adic
in eg al
ma ix
A
such
ha
H
=
AH'A'
.
Obse e
ha
his
algeb aic
no ion
o
genus
is
de ined
in
e ms
o
comple ion
whe eas
he
opological
one was
de ined
in
e ms
o localisa ion
.
This
di e ence
disappea s
i
only
non-singula
quad a ic
o ms
a e
conside ed,
o
hen
he
ing
Z
p o
p-adic
in ege s
may
be
eplaced
by
Z
(p
)
he
ing
o p-local in ege s in
he
de ini ion o
he
genus
o
an
in eg al
quad a ic
o m
(c
.
[2])
.
(Recall
ha
Z(
p
)
deno es
he
ing
o p-local
in ege s,
ha
is
he
ing
o
all
a ional
numbe s
wi h
denomina o
in
educed
o m
cop ime
o
p
.)
In
his
case he
simila i ies
be ween
he
wo
concep s
o
genus
a e
e en
mo e
sugges i e
.
Weakening
he
algeb aic
no ion o
genus
sligh ly
makes
he
connec ion
clea e
.
De ini ion
.
Le
A
a
commu a i e
ing o cha ac e is ic 0
.
Le
wo
in eg al
quad a ic
o ms
be
.
gi en,
one
wi h
he
ma iz
G,
he
o he
wi h
he
ma iz
H
in
some
basis
.
Then
he
wo
o ms
a e said o
be weakly
A-equi alen
i
he e
a e
an
in e ible
A-ma iz
A
and
m, a
uni in A,
wi h
mG
=
AHA'
.
(This
condi ion
is
ob iously
independen
o
he
choice
o
basis
.)
I
A
=
Z(p),
hen
we
speak
simply
o
weak
p-equi alen e
.
Two
non-singula
in eg al
quad a ic
o ms
a e
o he
same
weak
genus
i
hey
a e
weakly
Z(p)-equi alen
o
e e y
p ime
p
.
I
is
an immedia e
consequence
o
he
de ini ions
ha
he
weak
genus
o
an
in eg al
quad a ic
o m
is
an
in axian
o
i s
genus
.
Be o e
examining
examples
o
quad a ic
o ms
o
he
same
weak
genus,
i
should
be
obse ed
ha
we
may,
wi hou
loss
o
gene ali y,
es ic
ou sel es
o in eg al
compu a ions,
o
i
is
possible
o
"mul iply
up",
as
he
nex
lemma
makes
clea
.
Lemma
1
.

The
in eg al
quad a ic
o ms
G
and
H
a e
Z(
p
)
-equi alen¡
i
and
only
i
he e
a e
an
in eg al
ma iz
A
and
an
in ege
m
wi h
bo h
m
and
de (A)
cop ime
o
p,
such
ha
mG=AHA'
P oo
..
I
m'
is
a
uni
in
Z(
p
)
and
i
A'
is
an
in e ible
Z(p)-ma iz
such
ha
m'G=
A'H(A')',
hen
le
k
be
he
leas
common
mul iple
o
he
denomina o s
TOPOLOGICAL
AND
ALGEBRAIC
GENUS
OF
POLYHEDRA

327
o
he
en ies
in
A'
and
he
denomina o
o
m'
.
This
k
is
ce ainly
a
uni in
Z(n),
as
is
he
in ege
m
:=
k'm'
.
Mo eo e
A
:=
kA'
is
an
in eg al
ma ix
in e ible
o e
Z( ),
so
ha
i s
de e minan
(an
in ege )
is
a
uni
in
Z(n)
.
Bu
an
in ege
is
in e ible
in
Z(
p
)
i
and
only
i
i is
cop ime
o
p
.
Finally,
obse e
ha
mG
=
k
2
m'G
=
kA'Hk(A')`
=
AHA'
.
The
weak
genus
o
a
quad a ic
o m
is
equen ly
non- i ial
.
In
ac
he
diagonal
bina y
quad a ic
o ms
wi h
a
pai o
dis inc
p imes
on
he
diagonal
a e
always
o
he
same weak
genus,
,
some imes
(bu
no
always)
o
he
same
genus,
bu
ne e
equi alen ,
as
he
nex
esul s
show
.
Theo em
2
.
Fo
each
pai
o
dis inc
p ime
numbe s
p
and
q,
he
in eg al
quad a ic
o ms
wi h ma ices
0

q
)

and

p

p
q
)
a e
o
¡he
sane
weak
genus,
bu
no
equi alen
.
P oo
..
Since
p(0

ql
-_

1

(0
pq)
(0

1
p

(p
o
)
i
o
he
wo
o ms
a e
weakly
Z(,)-equi alen
o
e e y
p ime
7~
p
.
Bu
q
(p
0

q)
-
(oq

0l/
(0

q)
(
1

p)
,
so
ha
he
wo
o ms
a e
weakly
Z(n)-equi alen
as well
.
The
wo
o ms
a e
equi alen
only
i
he
in eg al
ma ix equa ion
(
1

0
pq)
-
(c d)
(p0
q)
(ab
d)
has
á
solu ion
.
This
is
only
he
case
i
he
equa ion
pa
2
+
qb
2
=
1
has
in eg al
solu ions,
which
i
clea ly
ne e
does
.

328

I
.
Boxon
Lemma
3
.
The
wo
qu¢d a ic
o ms
(0
41)
and
(0
82)
a e
o
he
same
genus
.
P oo
. .
The
equali y
C2
0)
_(-)C1

0)(
ó
77
0
41

'
7
0 82
-
shows
ha
he
wo
o ms
a e
Z(p)-equi alen
o
p
:~
7
and
he
equali y
i
ollows
ha
so
ha
Bu
o
any
in ege
x
4 -')
7
2
~2
0
-
109
iós


ó s
9
)
~
0

82)
(
10

109
0

09
shows
ha
he
wo
o ms
a e
Z(
7
)-equi alen
.
Lemma
4
.
The wo
qu¢d a ic
o ms
C~ 5)
and
(0 5)
a e
no
o
he
s¢me
genes
.
P oo
.Le kbe
an
in ege
and
A
he
2
x2
in eg al
ma ix
Ca
d)
Then
om
he
ma ix
equa ion
k2
(0
1
0
5)
-
(á d)
(0 5)
(ab
d)
k
2
-_
3a
2
+
5b2,
k
2
-
3a
2

(mod
5)
.
x
2
-
0,
l

(mod
5),
so ha
kmus be
di isible
by 5
.
In
o he
wo ds
C0
0
) and
( 0 15
)
TOPOLOGICAL
AND
ALGEBRAIC
GENUS
OF
POLYHEDRA

32
9
a e
no
5-equi alen
.
Tu ning
o
he
connec ion
be ween
he
opological
genus
o
he
mapping
cone
C
o
:
,son-1
V
S2n
and
he
algeb aic
genus
o
HM,
he
Hil on-Hop
quad a ic
o m
o
,
he
homo opy
commu a i e
diag am
induces
he
algeb aic
equa ion
S4n-1

~
S2n
9
deg(0)H(g)
=AMH( )A(sp)
and
V
and
cp
a e
p-equi alen es
i
and
only
i
bo h
deg(0)
and
de (A(cp))
a e
cop ime
o
p -
in
o he
wo ds,
i
and
only
i
H( )
and H(g)
a e
weakly
p-
equi alen
.
The
nex
heo em
summa ises
hese conside a ions
.
Theo em
5
.
The
mapping
ones
C
and
C
y
a e
o
he
same
genus
only
i
H( )
and
H(g)
a e o ¡he
same
weak
genus
.
Thus
he
weak
genus
o
he
Hil on-Hop
quad a ic
o m
H( )
is
an
in a ian
o
he
genus
o
C ,
as
is
he
genus
o he
suspension
o
C
.
In
ac ,
hese
wo
in a ian s
cha ac e ise
he
genus
o
C
in
he
case
ha
he
bouque
o
sphe es
V
S
2
n
consis s
o
p ecisely
one
sphe e,
o
in
ha
case
he
Hil on-Hop
quad a ic
o m
educes
o he
classical
Hop
in a ian
o
which
is
a
single
in ege
;
and wo
in ege s
-
in eg al
quad a ic
o ms
o
dimension
1
-
a e
o
he
sameweak
genus
i
and
only
i
hey
ag ee
up
o sign
.
Hence
he
Classi ica ion
Theo em
o
[1]
can
be e o mula ed
as
Theo em
6
.
Two
spaces
in
C,'
a e o
he
same
genus
i
and
only
i
(i)
hei
Hil on-Hop
quad a ic
o ms
a e
o ¡he
same
weak
genus,
and
(ii)
hei
suspensions
a e
o ¡he same
genus
.
This
e o mula ion
in i es
he
conjec u e
ha
he
esul
gene alises
o
he
case in
which
he
bouque
con ains
an
a bi a y
( ini e)
numbe
o
2n-sphe es
.
Conjec u e
.
Two
spaces
in
Cñ
a e
o
he
same
genus
i
and
only
i
(i)
hei
Hil on-Hop
quad a ic
o ms
a e o he
same
weak
genus,
and
(ii)
hei
suspensions
a e
o
he
same
genus
.
O
cou se he
"only
i '
pa o
he conjec u e
is
ce ainly
ue
.
33
0

I
.
BOKOR
Fo
K
=
0
he
conjec u e
is
ue
because
any
space
in
Cñ
is
homo opy
equi alen
o
S4n,
being
a
simply
connec ed
Moo e
space
wi h
he
app op i-
a e
homology
.
Theo em
6
asse s
he
conjec u e
o
be
ue
o
K
=
1
.
The
conjec u e
is
alsó
clea ly
ue
i
he
o sion
subg oup
o
7 4
n-1(S
2n
)
is
i ial,
o
hen
he
equa ion
in ol ing
he
o sion
componen s
imposes
no
addi ional
cons ain ,
and
e e y
K
x
K
in eg al
ma ix can
be
ealised
as
a
sel -map
o
VS'
n
, o anym>1
.
Ne e heless,
i
K
>_
2
and
i
he
o sion
subg oup
o
7 4n-1
(S2n)
is
non-
i ial,
hen
coun e examples
o
he conjec u e
can
be
sys ema ically
con-
s uc ed
.
Two
schema a
ollow
o
doing
so o
n1
{l,
2,4}
when
he
ke nel
o
he
suspension
map
E
:
7 4
.-1(V
S2n)
,
7 4n
(
V
S2n+1)
consis s p ecisely o
hose
classes
[ ]
whose
o sion
componen
is
i ial
.
Coun e example
1
.
I
he
o sion
subg oup
o
7 4n-1
(S
2
n)
i
s
non- i ial,
hen
he e
a e
mapping
cones
o
maps
S4n-1
-~
S2n
V
S2n
wi h
he
ollowing
p ope ies
.
(i)
Thei
Hil on-Hop
quad a ic
o ms
a e
o
he
same
weak
genus
.
(ii)
Thei
suspensions
a e
homo opy
equi alen
.
(iii)
They
a e
no
hemsel es
o he
same
genus
.
Coun e example
2
.
I
he
o sion
subg oup
o
7 4n-1
(S
2
n)
is
no
cyclic,
hen
he e
a e
mapping
cones
o
maps
S4n-1
,
S2n
V
S2n
wi h
he
ollowing
p ope ies
.
(i)
Thei
Hil on-Hop
quád a ic
o ms
a e
equi alen
.
(ii)
Thei
suspensions
a e
homo opy
equi alen
.
(iii)
They
a e
no
hemsel es
o he
same
genus
.
Cons uc ion
o
Coun e example
1
.
We
assume
ha
.T,
he
o sion
subg oup
o
1 4
n
-1
(S
2n
),
is
non- i ial
.
Le
p
be
a
p ime
di iso o
,
he
o de
o
T
.
Then
T
has
an
elemen
x o
o de
p
.
Le
qbe
any
p ime
numbe
o he
han
p
.
Finally
choose
,
g
:
S4n-1
-+
S2n
V
SU
wi h
he
ollowing p ope ies
:
2p
0
LLU)

0

2q)
and
E( )

G)
=
(
2

2pq

and E(g)
=
(0)
Then
Theo em
2
asse s
ha
H( )
and
H(g)
a e
o
he
same weak
genus,
e i ying
(i)
.
The
suspensions
a e
clea ly
homo opically
equi alen
-
in
ac
bo h
a e
ho-
mo opically
equi alen
o
CE

V
S
2
n+
1
-
which
es ablishes
(ii)
.
Now
conside
he
homo opy
commu a i e
diag am
S4n-1


)
V
S2n
S4n-1
V
S2n
TOPOLOGICAL
AND
ALGEBRAIC
GENUS
OF
POLYHEDRA

33
1
(whe e
z~
has
deg ee
m
and
A(~p)
=

a

b
c
d)
I
ollows
om
he
equa ion
ha
mH(g)
=
A(W)H( )(A(so)'
mpq
=
pcz
+
qdz,
so
ha
p
di ides
d
.
Mo eo e ,
he
equa ion
A(w)E
=
MEg
shows
ha
ex
=
0,
so
ha
p,
which
is
he
o de
o x, also
di ides
c
.
Bu
hen
p
di ides de (A(cp))
as
well,
so
ha
o
no
such
diag am
is
W
a
p-equi alence
.
This
es ablishes
(iii)
he eby
es ablishing
he
i s
amily
o
coun e examples
o
he conjec u e
.
Cons uc ion
o
Coun e example
2
.
Fo he
second
cons uc ion
we
conside
he
case
when
he
o sion
subg opup
T
o
7 4n_1(S
2n
)
is
no
cyclic
.
In
ha
case
i
has
a
subg oup
G
isomo phic
,
o
Z/pZ
®
Z/pZ
o
some
p ime p
We
iden i y
G
wi h
Z/pZ
®
Z/pZ
.
Choose
an
in ege
wi h
0
<
<
p
.
Then
he e
a e in ege s s
and
wi h
-
ps
=
-1
.
We
may
assume
wi hou
any
loss
o
gene ali y
ha
0
<
<
p,
o
(
+
kp)
-
sp
=
-
(s
-
k)p
.
F om
his
i
also ollows ha 0
<
s
<
p
.
Pu
Then
clea ly
(
E
GL(2
;
Z),
wi h
in e se
W i ing
he
elemen s
o
G
as
ows
means
ha
he
ows
o
(-' can
be
ie ued as
elemen s
o
G, and
hence
aken
o
ep esen
homo opy
classes
o
maps
Son-'
-b
Szn
.

The
i s
ow
co esponds
o
he
elemen
(- ,
0) o
G
and
he
second
o
(s,
- )
.

I ,
u he mo e,
he
elemen s
o
Gz
a e
w i en
as
columns
wi h
elemen s
o
G
as
en ies,
hen
we
may
ega d
(-'
as
an
elemen
o
Gz
and
so
as
ep esen ing
a
homo opy
class
o
maps
Son-'
,
s2n
V
SI,
.
pu
y
:_
(o
.
Then
y
is
he
2
x
2 uni
ma ix
1
iewed
as
an
elemen
o
Gz
and
( de ines
an
au omo phism
o
Gz
mapping
x o
y
.
Now
Choose
,
g
:
son-i
-,
szn V
Szn
wi h
he
ollowing p ope ies
H(
)
=
(
0

2
)

and
E(
)
_
(
s

p
)
H(9)
_
(~

~)
and
E(9)
=
(

~)