Publicacions
Ma emá iques,
Vol 36
(1992),
47-50
.
A
bs ac
TORSION
UNITS
IN
GROUP
RINGS
VIKAS
BIST
Le
U(RG)
be
he
uni
g oup
o
he
g oup
ing
RG
.
In
his
pape
we
s udy g oup
ings
RG
whose
suppo
elemen s
o
e e y
o sion
uni
a e
o sion,
whe e
R
is
ei he
he
ing o in ege s
7L
o
a
ield
K
.
Le
R
be
a
commu a i e
ing
wi h
iden i y,
G
be
a
g oup
and
U(RG)
be
he
g oup
o
uni s o
he
g oup
ing
RG
.
Deno e by T(G),
he
se
o
o sions
elemen s
o
G
.
I is
p o ed
in
[2],
ha
i
T(U(7LG))
is
a
subg oup,
hen
T(U(3G))
=
±T(G)
.
In
his
no e
we
s udy
g oup
ings
RG
whose
suppo
o
e e y
o sion
uni
is
in
T(G)
.
Theo em
1
.
Le
R
be
an
in eg al
domain,
F
be
i s
quo ien
ield
and
G
be
a
non
o sion
g oup
.
I
he
suppo
o
e e y
o sion
uni
o
RG
is
in
T(G),
hen
T(G)
is
a
subg oup
wi h
e e y
subg oup o
T(G)
no mal
in
G
and
e e y
idempo en
o
FT(G)
cen al in
FG
.
P oo
..
Le
E
T
(G)
be
o
o de
n and
le
x
E
G T
(G)
.

Then
a
=
+
(1
-
)x(1
-1-
+
.
.
.
+
n
-1
)
E
U(RG)
andan=
1
.
Since
supp(o)
C_
T(G)
;
x
=
x
k o
some
k
=
1,
2,
. .
.,
n -
1,
hus
x
-1 x
=
x
E
( )
.
+3
i
x
E
G T(G),
hen
x
E
NG(( )),
whe e
NG(( ))
is
he
no malize
o
( )
in
G
.
I
y
E
T(G)
and x
E
G T(G),
hen
x
E
NG((y))
.
Since
NG((y))ICG(y)
is
ini e,
so
x'
c
CG(y)
o
some
posi i e in ege
m
.
Now
(xy)
m
=
xmYX
--l
YX
--2
. . .
yx
y
and
as
y'
E
(x),
so (xy)
TnI
=
xmk,
whe e
k
is
he
o de
o
y
.
Hence
xy
is
o
in ini e
o de
.
Thus
xy
E
NG(( ))
and
so
y
E
NG(( ))
.
Hence
( ) is
no mal
in
G
o
e e y
E
T(G)
.
48

V
.
BIST
Le
e
be
an
idempo en
in
FT(G)
and x
E G T(G)
.
The e
exis s
E
R
such
ha
ex(1
-
e)
E
RT(G)
and
so
1
+
ex(1
-
e)
E
U(RG)
wi h
(1
+
ex(1
-
e))
-1
=
1
-
ex(1
-
e)
.
Now
o
any
E
T(G),
(1
-
ex(1
-
e)) (1
+
ex(1
-
e))
E
T(URG)
and
+
xb
-x
2
0, whe e
b
=
(
x
e
x
(1
-
e)
-
e
x
(1
-
e)
)
and
O
=
2e
x2
(1
-
e
x
)
( e)x(1
-
e),
6,
O
E
RT(G)
.
Since
,3
E
TU(RG)
and
T(U(RG))
C
(RT(G)),
so
supp(0)
C
T(G)
.
Thus,
b
=
0
and
O
=
0
.
Now
xS
=
0
implies
ha
ex(1-e)
=
ex(1-e)
.
Hence
ex(1
-
e)
commu es
wi h
e e y
elemen
o
RT(G)
.
Thus
ex(1
-
e)
=
0
.
Simila ly,
we
ha e
(1
-
e)xe
=
0
.
Thus
i
ollows
ha
ex
=
e
o
e e y
x
EG T(G)
.
Now
i
y
E
T
(G)
and x
E
G T
(G),
hen
xy
is
also
o
in ini e
o de
.
So
ey
=
(ex)y
=
ex!'
=
e
.
Thus
e
is
cen al
in
RG
.
This
p o es he
esul
.
By
i ue
o
he
abo e
heo em
he
p oblem
hus
educes
o
de e mine
RG
such
ha
T(U(RG))
C_
U(RT(G))
.
We
now
assume
ha
R
is
ei he
he
ing
o
in ege s
7L
o
a
ield
K
.
Fo
he
in eg al
g oup
ings
7G,
we
ha e
he
ollowing
si ua ion
.
Theo em
2
.
Le
G
be
a
non o sion
g oup
such
ha
T(G)
is
a
sub-
g oup
and
ha
G/T(G)
be
igh
o de ed
.
Then,
he
ollowing
condi ions
a e
equi alen
:
(1)
TU(7LG)
C
U(7LT(G))
(2)
T(G)
is
ei he
abelian
o a
Hamil onian
g oup
such
ha
i
T(G)
is
nonabelian,
a
E
T(G),
o
odd
o de
n,
hen
he
mul iplica i e
o de
o
,2
in
7L
n
is
an odd
numbe
(3)
U(7ZG)
=
U(ZZT(G))G
.
P oo
.
(1)
implies
(2)
.
I
T(G)
is
non
abelian,
hen
by
Theo em
1,
T
(G)
=A
x
E
x
Kg,
whe e
A
is
abelian
wi h
e e y
elemen
o
odd
o de ,
E
is
elemen a y
abelian
2-g oup
and
K8
is
he
Qua e nion
g oup
o o de
8
.
Le
a
E
A
be
o ode
n
.
Then
by
[4,
11
.2 .6]
(q((a))
x
Ks)
=
(q(a))Ks
=
®
YI
Q(~d)Ks-
dln
Also
C¿( ,,)Ks
-
Q(bn)
®

®
Q(~n)
®
S,
whe e
S
is
ei he
a
di ision
ing
o
M2(Q(~n))
.
By
Theo em
1,
e e y
idempo en
o
QT(G)
TORSION
UNITS
IN
GROUPS
RINGS
is
cen al
in
q(G),
so
Q(~,)Ks
has
no
noncen al
idempo en s
.
Thus
S
is
a
di ision
ing
and
he e o e,
Q(~

,)Ks
has
no
nonze o
nilpo en
elemen s
.
By
[4,
VIT13]
a
2
+
b2
+
c
2
=
0
has
no
nonze o
solu ion
in
Q(~

,)
and
by
[4,
VI
.1
.15],
his
happens
p o ided
he
mul iplica i e
o de
o 2
modulo
n
is
odd
.
(2)
implies
(3)
is
by
[1]
.
(3)
implies
(1),
ollows
om
an
easy
obse a ion
ha
U(W)/U(ZTCG))=
G/T(G)
.
E
Finally
o
g oup
algeb as
we
ha e
he
ollowing
heo em
.
He e
K
*
G
deno es
he
c ossed
p oduc
o
G
o e
K
.
Theo em
3
.
Le
K
be
a
ield
o
cha ac e is ic
p
>
0,
G
be
a
non
o sion
g oup
such
ha
T(G)
is
a subg oup
and
G/T(G)
be
igh
o de ed
.
Fu he
le
G
be
such
ha
o e e y
ini ely
gene a ed subg oup
H
o
G,
T(H)
is
cni e
.
Then
T(U(KG))
C_
U(KT(G))
i
and
only
i
T(G)
is
abelian
g oup
ha ing
no
p-elemen s
and
e e y
idempo en
o
KT(G)
is
cen al
in
KG
.
P oo
..
Suppose
ha
T(U(KG))
C
U(KT(G))
.
Then
by
Theo em
1,
e e y
subg oup
o
T
(G)
is
no mal
in
G
wi h
e e y
idempo en
o
KT
(G)
cen al
in
KG
.
I
cha
K
=
p
>
0and
E
T(G)
wi h
o( )
=
p,
hen
as
( )
is
no mal
in
G,
so
IG
:
Cc
( )
i

<
oo
.

Siüce
G
is
non
o sion,
he e
exis s
an
elemen
x
o
in ini e
o de
in
CG( )
.

Then
(1
+
x(1
-
))
=
1
and
1
+
x(1
-
)
~
KT(G)
.
Hence
T(G)
has
no p-elemen s
.
Finally
i
T(G)
is
non
abelian,
hen
he
Qua e nion
g oup,
Ks
C
T(G)
and
p
:,A
2
as
T
(G) has
no
p-elemen s
.
So
ZpKs
=
Zp
®
7L
p
ED
Z
I,
ED
7Z
p
m
M2
(7L,,),
con ains
anon
cen al
idempo en
.
Hence
T(G)
is
abelian
.
Fo
he
con e se,
we
may
assume
ha
G
is
ini ely
gene a ed
and
so
T(G)
is
ini e
.
Now
KT
(G)
=
Fi,
a
di ec
sum
o
ields,
since
T(G)
is
ini e
abelian
and p
does
no
di ide
jT(G)j
.
I
is
gi en
ha
e e y
idempo en
o
KT(G)
is
cen al
in
KG
.
Hence
KG
=
KT
(G)
*
G/T
(G)

F
i
*
G/T
(G)
i=I
49
5 0

V
.
BIST
and
so
This
p o es
he
heo em
.
Since
G/T(G)
is
igh
o de ed,
;
by
[4,
VI
.
1
.61
U
(F
i
*G/T(G))
has
only
i ial
uni s
and
so
T(U(KG))
C
D T(U(Fi
*
G/T(G)))
=
D T(U(F
i
))
C
U(K(T(G))
.
By
Theo em
3
and
[3]
we
ha e
U(KG)
=D
lU(Fi
*
G/T(G))
.
Co olla y
4
.
Le
K
be
a
ield
o
cha ac e is ic
p
and
G
be
non o -
sion
nilpo en
o
FC-g oup
ha ing
no
p-elemen s
.
Then
T(U(KG))
C
U(KT(G))
i
and
onlg
i
T(U(KG))
is
a
subg oup
.
Re e en es
1
.

A
.A
.
BOVDI,
Cons uc ion
o
an
in eg al
g oup
ing
wi h
i ial
elemen s
o
ini e
o de ,
Sibi sk
.
Ma
.
Zh
.
21
(1980),-28-37-
2
.

C
.P
.
MILICS,
G oup
whose
o sion
uni s
o m
a
subg oup,
P oc
.
Ame
.
Ma h
.
Soc
.
81
(1981),
172-174
.
3
.

C
.P
.
MILICS,
G oup
whose
o sion
uni s
o m
a
subg oup
II,
Comm
.
Algeb a
9
(1981),
699-712
.
4
. ,
S
.K
.
SCIIGAL,
"Topics
in
g oups
ings,"
Ma cel
Dekke ,
New
Yo k,
1978
.
Depa men
o
Ma hema ics
Punjab
Uni e si y
Chandiga h
-
160014
INDIA
Rebu
el
2'1
de No eTn,b e de
1990