INVARIANT SUBSETS OF PLQ( m ) UNDER THE ACTION OF = 〈 , : = 1)

Abstract

Let n k m2= . Then ( {=) ,,: ,(, =), 1} 2 2*cc a n Z a c a n
ca
c
a n
Q n
−
∈
+ −
is a G -subset of Q( m \) Q
where = 〈 , : = 1= 〉
2 3 G x y x y . In this paper we find proper M -subsets of
)) ( )
9
\) (
9
( {=) ( :) 0( 3)} ( =) (
*** * ~* * *** *** Q n
n
Q
n
Q n c mod or Q n Q
c
a n
Q n ∈ ≡ ∪
+
according as n ≡/ 0(mod 9) or n ≡ 0(mod 9) and ( 9 (=) ( \) ( )) ( 9 )
~* * *** *** Q n Q n Q n ∪ Q n for all
n which are invariant subsets of Q( m \) Q under the action of = 〈 , : = 1= 〉
2 6 M x y x y . Specifically we show
that (9 =) (9 )
~*
oM p oG p for all prime p
, where (9 )
~*
o p M
denotes the number of M -orbits of ( 9 )
* Q p
:
and
o (9 p) G
denotes the number of G -orbits of ( 9 )
* Q p . Also we prove that =)( )(
***
oM p oG p if p ≡ 1(mod 3)
where )(
***
o p M
denotes the number of M -orbits of ( )
*** Q p .
AMS Mathematics subject classification (2000): 05C25, 11E04, 20G15