# Some non-maximal arithmetic groups

Some non-maximal arithmetic groups
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Let k be a non-finite Dedekind domain, and σ be the ring of its integers. We shall assume that the ring R = σ- 2 is finite. Let us denote by Mn k resp. Mnσ the ring of all n by n matrices with entries in k resp. in σ, and Gln k its group of units.We denote by sln k the subgroup of Gln k whose elements g have determinant, det g, equal to one. Let H ε Mn σ be a symmetric matrix, i.e., H = tH where tH denotes the transpose matrix of H. We let G = SO H = { g ε Sln k l tgHg = H }, and we let Gσ = G∩Mn σ. We want to exhibit certain H for which Gσ is not maxinal in G, in the sense that there exist a subgroup Δ contains Gσ properly and Δ : Gσ is finite.

Tipo de documento: Artículo - Article

Palabras clave: Teoría de los números, grupos discontinuos, grupos aritméticos

Source: http://www.bdigital.unal.edu.co

## Teaser

Revista Colombiana de Matematicas
Volumen II,1968, pags.

21- 28
SOME NON-MAXIMAL ARITHMETIC GROUPS
by
Nelo D.

ALLAN
Let
k
be a non-finite Dedekind domain, and ~
be
the ring of its integers.

We shall assume that the ring
R =)r-(2)
Mn()r))
in
is finite.

Let us denote by
the ring of all
k
(resp.

in):f), and
se n (k)
We denote by
ments
t
by
n
ox.

n (k)
its group of units.
n
H
denotes the transpose matrix of
SO(H)
=
{g
maximal in
G,
of
r,
H
S~n (k) - tgHg
E
H
We want to exhibit certain
group A
O£ (k)
the subgroup of
n
whose ele-
e~ual to one.
Let
H
where
be a symmetric matrix, i.e.,
(Jf)
(resp.
n
matrices with entries
have determinant, det g,
g
HEM
n
M (k)
tH
=
H.

We let
G
and we let Off
for
which
Gff
=
=
GnMn (JY) •
is not
in the sense that there exists a sub-
G
such that
t::,.
contains
Off
properly and
[e:.:GhJis finite.
1.
E!~11~1E~Ei~~
Let L
shall denote by
all the
L .
be an order in
Mn(k); we
the fractional ideal generated by
lJ
(i,j)-entries of all the elements of
,
L
we
shall write
L
We shall say that
dule
L
L
is a direct summand if as an;.r-mo-
is a direct sum of
the units of
nn
L .

e .
l.J l.J
where
e .
l.J
are
M (k).
n
21
It
in
]V!
n
is well known that
(k)
are conjugate
mmands and
tl
L
nJ
L
I
, i, j
L ,J
=
nn
1
to the ones -hich
=;,t:,
for
n,
in our case the maximal orders
I
i,j
n,
are direct
and
some fractional
su-
=,a-l
L.
ln
·U of
ideal
k,
i .e.

,
L
L
If
is
pansion
one of such orders,
g-1 ,
of
a group.
g
then by 100kinB at the ex-
Sen (k )
E
Consequently
if
j
11 SEn(k) is
we see that
L
sf n (k),
!J.=
Gc
then
n
G
is
L
a group.
For our purposes
1~M~~1.

If
surable
both
to
R
we consider
=~-~
follow
The index
r~ff:
the order
of the group
6 (it)]
(v .

,)
lJ
g
and
-1 ,;
g
v , ,::::0
lJ
-1
E g V,
M (.e;).
only finitely
ly finitely
611
follows
22
all
that
finite.
tg
=...