Abstract
Let C
0
= C ∪ {∞} be the extended complex plane and
M =
x, y : x
2 = y
6 = 1®
, where x(z) = −1
3z
and y(z) = −1
3(z+1) are
the linear fractional transformations from C
0 → C
0
. Let m be a squarefree positive integer. Then Q∗
(
√
n) = {
a+
√
n
c
: a, c 6= 0, b =
a
2−n
c
∈
Z and (a, b, c) = 1} where n = k
2m, is a proper subset of Q(
√
m)
for all k ∈ N. For non-square n = 3h Qr
i=1 p
ki
i
, it was proved in an
earlier paper by the same authors that the set Q
000 (
√
n) = {
α
t
: α ∈
Q∗
(
√
n), t = 1, 3} is M-set ∀ h ≥ 0 whereas if h = 0 or 1, then
Q∗∗∗√
n) = {
a+
√
n
c
:
a+
√
n
c
∈ Q∗
(
√
n) and 3 | c} is an M-subset
of Q
000 (
√
n) = Q∗
(
√
n) ∪ Q∗∗∗(
√
9n). In this paper we prove that if
h ≥ 2, then Q
000 (
√
n) = (Q∗
(
pn
9
)\Q∗∗∗(
pn
9
))∪Q∗
(
√
n)∪Q∗∗∗(
√
9n)
and also determine its proper M-subsets. In particular Q(
√
m) \ Q =
∪Q
000 (
√
k
2m) for all k ∈ N.
M. Aslam Malik, S. M Husnine, Abdul Majeed. (2012) Action of the mobius group ¨ M = hx, y : x 2 = y 6 = 1i on certain real quadratic fields, Punjab University Journal of Mathematics, Volume 44, Issue 1.
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