Skopje, Makedonija ABOUT CHARACTERS ON VILENKIN ...

Matematiqki Bilten ISSN 0351-336X
32 (LVIII)
2008 (35-46)
Skopje, Makedonija
ABOUT CHARACTERS ON VILENKIN GROUPS
MEDO PEPIĆ
Abstract. In this paper we proved that Lemma [7,p.727] is not true and that
the following Theorem holds: Let G be Vilenkin group. Let k ∈ [m n,m n+1 )
∑
be natural number given by k =
n k j m j ,1≤ k n

36 MEDO PEPIĆ
where G j = {0, 1} is a cyclic group of the second order for all j ∈ N, equipped by
discrete topology, with the component adding (note that adding in each component
is done by module 2). It’s direct generalization is group
∞∏
G = Z nk ,
k=0
where Z nk := {0, 1, 2,...,n k − 1} ,n k ≥ 2, is a cyclic group of order n k (k ∈ N 0 )
equipped by discrete topology.
Arbitrary x ∈ G has the unique representation in the following form
∞∑
x = x j g j ,x j ∈{0, 1, 2,...,p j+1 − 1} (3)
j=0
where g j ∈ G j G j+1 are previously arbitrary chosen and fixed, and for all j ∈ N 0
holds
{
}
∞∑
G j = x ∈ G : x s g s ,x s =0, for 0 ≤ ss) ∧n j(n) ≠0,then

ABOUT CHARACTERS ON VILENKIN GROUPS 37
j(n)
∑
n = n j m j ,n j ∈{0, 1, 2,...,p j+1 − 1}∧1 ≤ n j(n)

38 MEDO PEPIĆ
Then all the elements of the subgroup Γ j+1 have been ordered and by induction
all the elements of Γ. In this kind of ordering, for n given by (5), obviously holds
∞∏
χ n = χ nj
m j
, 0 ≤ n j

ABOUT CHARACTERS ON VILENKIN GROUPS 39
of the corresponding coefficients. Addition of elements x =(x j ) j∈N0 and y =
(y j ) j∈N0 in G we denote by + and define it by component addition. Component
addition for arbitrary coordinate j ∈ N 0 is addition mod p j+1 .
For arbitrary n ∈ N given by (5) and arbitrary x ∈ g given by (3) we have
χ n (x) =
∞∏
j=0
χ n j
m j
(x), 0 ≤ n j

40 MEDO PEPIĆ
(19) becomes
χ n (x) =
∞∏
j=0
r n j
j (x), 0 ≤ n j

ABOUT CHARACTERS ON VILENKIN GROUPS 41
d) Γ 0 = {χ 0 }∧Γ 1 = {χ 0 ,χ 1 }∧Γ 2 = {χ 0 ,χ 1 ,χ 2 ,χ 3 ,χ 4 ,χ 5 }∧Γ 3 = {χ j :0≤ j

42 MEDO PEPIĆ
e) For n ∈ N given by (5) and x ∈ G given by (3) holds
⎛ (
)
∞∑ k j
j∑
⎞
χ n (x) =exp⎝2πi
· x s m s
⎠ .
m j+1
j=0
s=0
Proof. (Lemma 1).
a) By the facts that r n ∈ Γ n+1 Γ n and Γ n+1 Γ n is a cyclic group of prime
order p n+1 follows r p n+1
n ∈ Γ n Γ n−1 . Therefrom and using the fact that
Γ n Γ n−1 is a cyclic group of prime order p n follows r p n+1.p n
n ∈ Γ n−1 Γ n−2 .
By continuing reasoning describe above we conclude that for each 0 ≤ s ≤ n
holds rn
pn+1.pn....ps+1 ∈ Γ s Γ s−1 . This proves assertion a) since obviously holds
(∀s ∈{0, 1, 2,...,n})p n+1 .p n ...p s+1 = m n+1
m s
.
Let us notice, that from (21) follows
r p n+1.p n ...p s+1
n = r p s+1
s = r s−1 .
b) If s>n,then r n ∈ Γ s and therefrom r n (x s g s )=1sincex s g s ∈ Γ s Γ s+1 .Let
us arbitrarily choose s ∈{0, 1, 2,...,n}. Then x s g s ∈ Γ s Γ s+1 , for all 1 ≤
m n+1
ms
x s

ABOUT CHARACTERS ON VILENKIN GROUPS 43
χ k (x) =
n∏
j=0
e) By (5) ∧ (18 ∗ ) ∧ c) wehave
χ n (x) =
r kj
j
n (x) = ∏
k j ·
exp ⎜
⎝ 2πi
j=0
j=0
⎛
⎞
j∑
x s m s
⎟
⎠
s=0
m j+1
⎛ (
)
n∑ k j
j∑
⎞
= exp⎝2πi
· x s m s
⎠ .
m j+1
∞∏
j=0
r kj
j
s=0
∞ (x) = ∏
k j ·
exp ⎜
⎝ 2πi
j=0
j=0
⎛
⎞
j∑
x s m s
⎟
⎠
s=0
m j+1
⎛ (
)
∞∑ k j
j∑
⎞
= exp⎝2πi
· x s m s
⎠ .
m j+1
s=0
□
Proof. (Theorem 1). Let G be given Vilenkin group. Let k ∈ [m n ,m n+1 )be
natural number given by
k =
n∑
k j m j , 1 ≤ k n

44 MEDO PEPIĆ
k =
n∑
b j m j , 1 ≤ b n

ABOUT CHARACTERS ON VILENKIN GROUPS 45
x =(0, 0,...,0,g n−1 , 0,...).
For such x we have x n−1 =1∧ x s =0foralls ≠ n − 1, so in that case (29)
implies
k n−1 + b n−1
m n
which is according to (30), equivalent to
k n−1 + b n−1
p n
+ 1
p n
∈ N 0 , i.e.
· m n−1 + k n + b n
m n+1
· m n−1 ∈ N 0 ,
k n−1 + b n−1 +1
p n
∈ N 0 .
If we take account that 0 ≤ k n−1

46 MEDO PEPIĆ
Substituing (33) in (26) we obtain
n−1
∑
k =(p n+1 − k n )m n + (p j+1 − k j − 1)m j = m n+1 − k + m n − 1.
j=0
Now it is easy to check, that for k determined in a such way, really,
holds true. Namelly,
⎛
n∏
χ k · χ k
= ⎝
j=0
r kj
j
⎞ ⎛
n∏
⎠ · ⎝
χ k · χ k
= χ 0
j=0
r bj
j
⎞
⎠ = r p n+1
n ·
⎛
n−1
∏
⎝
j=0
r pj+1−1
j
⎞
⎠ = χ 0
(according to (23) and (21).
□
References
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Acta Sci. Math. (Szeged) 69 (2003), 687-732.
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Department of Mathematics, University of Sarajevo, Zmaja od Bosne 33-35, Bosnia
and Herzegovina,
E-mail address: mpepic@lol.ba