Open Journal of Discrete Mathematics, 2011, 1, 136-138
doi:10.4236/ojdm.2011.13017 Published Online October 2011 (http://www.SciRP.org/journal/ojdm)
Copyright © 2011 SciRes. OJDM
An Application of Cyclotomic Polynomial to
Factorization of Finite Abelian Groups*
Khalid Amin
University of Bahrain, Department of Mathematics, Sakhir, Kingdom of Bahrain
E-mail: kamin@sci.uob.bh
Received June 21, 2011; revised July 25, 2011; accepted August 21, 2011
Abstract
A finite abelian group Gis said to have the Hajós-k-property (k>1) if from any decomposition
of into a direct product of its subsets, it follows that one of these subsetsis periodic,
meaning that there exists a nonidentity element g in G such that gAi = Ai. Using some properties of the
cyclotomic polynomials, we will show that the cyclic groups of orders pα and groups of type (p2, q2), where p
and q are primes have this property. We also include a partial result about groups of type (pα, qβ), where p
and q are distinct primes and α, β are integers 1.
K
AAAG ...
21
Gi
A
Keywords: Factorization of Finite Abelian Groups, Hajós Property
1. Introduction
Let be a finite abelian group with identity element
. If is a direct product of cyclic groups of orders
12 , we say that is of type.
If 12
G
G
,,mm
e
,
r
m
,
,,
G
12
,,,
r
mm m
k
A
AA are subsets of G such that each ele-
ment
g
of G can be expressed in a unique way as
=
g
k, where ii
, we write
12 k and say that we have a factorization of
. If in addition each contains the identity element ,
we say that we have a normalized factorization of .
We will use
12
aa
a
A
aA
G
G
=,AGA
G
i
A
to denote the number of elements of
i
A
. Similarly
g
will donate the order of the element
g
of G. A subset
A
of G is called periodic if there
is a non-identity element
g
in such that G=
g
AA.
The topic of factorizations of abelian groups arose
when Hajós [2] solved a conjecture by H. Minkowski [3]
concerning lattice tiling after transforming it into a theo-
rem about finite abelian groups. For reference, we state
this theorm below:
If is a factorization of a finite abe-
lian group , where each of the subsests is of the form
12
=,
k
GAAA
G
,,, r
eaa a
G
, then at least one of these subsets is a
subgroup of .
L. Rédei [4] generalized this to:
If is a factorization of a finite abe-
lian group , where each of the subsets has a prime
number of elements and contains the identity e, then at
least one of these subsets is a subgroup of .
12
=,
k
GAAA
G
G
A. Sands [5] classified groups with Hajós-2-proprty
which we list below:
 


 


22 2
32 2
2
22
,, ,, ,,,,
,,,2,2,2, ,2,2, ,2,2
,2,2 ,,3,3 , 3,3 ,2
2,2 ,,
qp qp qrpqr
pp p
p
pp

,,
2,2,
,2
s
p

2,2
,
p
pq
where p, q, r and s are primes and 1
is an integer.
2. Preliminaries
Let be a cyclic group of order n, with generator g. G
Let us write. Similarly, for a subset
1
0
n
i
i
gG
,,
r
g
12
g

=,Ag of G, we write
0
=
r
i
i
A
g
2
r
i
g
.
Then we can write
12
11 1
=... ,
ii
rr
k
ii
AA Agg
 





12
i
Where multiplication is carried out in the group ring
. Thus, when multiplication is carried out we re-
gard
)(GZ
i
A
as polynomials in g provided that addition of
*Mathematics Subject Classification: 20K01
K. AMIN 137
the indices is carried out module n. i.e. polynomials are
multiplied .

1
n
mod g
)...()( 21 xAxA

1x
Now, if we replace g by x and write
then from the relation , then we get:
i
i
r
i
ixxA
1
)(
12
=,
k
GAAA
)1mod()()(  n
kxxAxG
As is a factor of, it follows that
each irreducible divisor of
1n
x
1
n
x
1
nx

1
x will divide
one of the polynomials i
A
x. These irreducible poly-
nomials are the cyclotomic polynomials whose roots are
the d-th primitive roots of unity where dn and. >1d
At some stage in this work, we shall need the follow-
ing facts about the cyclotomic polynomials.
1) The n-th cyclotomic polynomial is usually denoted
by
n
x
and is given by:


=1
d
n
d
n

n
dn
xx





2) The
x
s
have integer coefficients i.e.
n
x
()
Z
xand they are irreducible and relatively prime.
Slightly modifying the notation of De Bruijn [1], we
also define for a divisor d of n, the polynomial


21
,
1
==1
1
nn
nd
dd
nd n
d
x
xxx
x

n
d
x
3. Results
Before we embark on showing our results, we must men-
tion that all factorizations can be assumed to be normal-
ized, for if is a factorization of G,
then since each
12
=,
k
GAA A
i
A
is non-empty, there exists an ele-
ment ai is Ai, 1 Multiplying G by
we get that
Gg which is
clearly normalized.
.ik
11
==Ga
12
=,
k
gaa
1
, ,
kk
aA
,
1
a
11
22
AaA

Theorem 3.1 Let p be a prime. If G is a cyclic group
of order =np
, then G has the Hajos-k-proprty, for all
k, 1< k
.
Proof Let G be generated by g and consider the fac-
torization 12 of G. Our previous discus-
sion leads to the following congruence relation:
=,
k
GAAA
 

12 1
n
k
GxAxAxAxmodx.
Now,

n
x
divides some()
i
A
x. But
 
1
1
==
1
p
npp
x
xx
x

.
Thus,
1
1
1
p
p
x
x
divides
i
A
x
1
. It follows that is periodic
with period
i
A
p
g
As an illustration, consider a cyclic group =Gg
of order Let
3
8=2.
45
=,,,
A
egg g and
2
=,Beg
31
.
Then it is easy to verify that AB = G is a factorization of
G and that A is periodic with period42
=
g
g.
We shall use the following theorem by De Bruijn [1]
in showing our next result.
If =npq
,1
, where are distinct primes and ,pq
,
F
xZx and

n
x
divides
F
x,
then

,
nq

,
=np
F
xgx xhxx for some po-
lynomials
,
g
x
hx Zx.
Theorem 3.2 If G is of type
, where p and q
are distinct primes, then G has the Hajos-k-property, for
all k,
2
,pq
2
1< 4k
.
Proof Consider a factorization 12 of
. The case is true by Redei's theorem. The
case , is true by Redei theorem and Theorem 2 of
Sands [5]. Thus, we only need detail the case
Say 12
=,
k
GAAA
k
G=4k
=3k
=G
=2.
A
A in which both factors contain
elements. Again by our previous discussion, we obtain the
relation
pq
12
()() ()mod(1),
n
GxAxAxx
where It follows that
22
=npq.

n
x
divides
12
()
A
xA x . Since 1
A
and 2
A
contain the same
number of elements, we may and shall assume that
n
x
divides
1
A
x. Then, by De Bruijn’s result
above, we get that

1,
=.
np nq
,
A
xfxxgxx

Therefore,
 
1
11
=
11
nn
nn
pq
xx
Ax fxgx
x
x


.
Now:

11= =1(1)
A
pq pfqg
Therefore, either
1=
f
q and, or (1) = 0g
1=f
and
0(1) =
g
p. In the first case, , and

=0gx
1
1
n
np
x
x
divides
1
A
x in which case 1
A
is periodic
with period np
g
. In the second case, and

=0fx
1
1
n
nq
x
x
divides
1
A
x and so 1
A
is periodic with pe-
riod np
g
.
Corollary 3.1 If G is of type

,pq
, where p and q
are distinct primes, and ,
are integers , then G 1
Copyright © 2011 SciRes. OJDM
K. AMIN
Copyright © 2011 SciRes. OJDM
138
has the Hajós-k-property, where

=2k

.
4. References
[1] N. G. De Bruijn, “On the Factorization of Finite Cyclic
Groups,” Indagationes Mathematicae, Vol. 15, No.4,
1953, pp. 370-377.
[2] G. Hajos, “Uber Einfache und Mehrfaache Bedekung des
n-Dimensionales Raumes Mit Einem Wurfelgitter,” Ma-
thematics Zeitschrift, Vol. 47, No. 1, 1942, pp. 427-467.
doi:10.1007/BF01180974
[3] H. Minkowski, “Diophantische Approximationen,” Teu-
ner, Leipzig, 1907.
[4] L. Redei, “Ein Beitrag Zum Problem Der Faktorisation
Von Endlichen Abelschen Gruppen,” Acta Mathematics
Hungarica, Vol. 1, No. 2-4, 1950, pp. 197-207.
doi:10.1007/BF02021312
[5] A. Sands, “Factorization of Finite Abelian Groups,” Acta
Mathematics Hungarica, Vol. 13, No. 1-2, 1962, pp. 153-
169. doi:10.1007/BF02033634