Internat. J. Math. & Math. Sci.
VOL. 12 NO. 2 (1989) 267-283
267
ON FACTORIZATIONS OF FINITE ABELIAN GROUPS WHICH ADMIT
REPLACEMENT OF A Z-SET BY A SUBGROUP
EVELYN E. OBAID
Department of Mathematics and Computer
Science
San Jose State University
San Jose, CA 95192
(Received October 20, 1987)
A subset A of a finite additive abelian group G is a Z-set if for all
Z.
A for all n
A, na
ABSTRACT.
a
The purpose of this paper is to prove that for a special class of finite abelian
groups, whenever the factorization G
the series G
K
m
K
Kn
2
the factorization G
A
B, where A and B are Z-sets, arises from
<0> then there exist subgroups S and T such that
T also arises from this series.
S
This result is obtained
through the introduction of two new concepts: a series admits replacement and the
extendability of a subgroup.
A generalization of a result of L.
Fuchs is given
which enables establishment of a necessary and sufficient condition for extendability.
This condition is used to show that certain series for finite abelian p-groups admit
replacement.
KEY WORDS AND PHRASES.
Finite abelian group, factorization, Z-set.
1980 AMS SUBJECT CLASSIFICATION CODES. 20K01, 20K25.
I.
INTRODUCTION.
Let G be a finite abelian additive group and let A and B be subsets of G. If
a + b, where a A,
every element g G can be uniquely represented in the form g
bB, then we write G
to be a Z-set if for all
A
B and call this a factorization of G.
A subset A
is said
aA, nasa for all nZ.
A.D. Sands [I] gave a method which yields all factorizations of a finite abelian
His method corrects one given previously by G. Hajos [2].
good group.
Our main purpose is to prove that for a special class of finite abelian groups,
whenever the factorization G
G
K
K2
Kn
K
n
A
Kn
S
B, where A and B are Z-sets, arises from the series
<0> (see [3]) then there exist subgroups S and
T such that factorization G
T also arises from this series.
In order to achieve this result we introduce two new concepts: a series admits
replacement and the extendability of a subgroup. We prove a generalization of a
result of L. Fuchs [4] which enables us to derive a necessary and sufficient condition
for extendability.
This condition is used to show that certain series for finite
abelian p-groups admit replacement.
268
E.E. OBAID
PRELIMINARIES.
2.
We shall use the term "Z-factorization" when referring to a factorization of
A H B, where A and B are Z-sets
the form G
Our first two lemmas can be readily verified.
LEMMA I. Let G S H A, where S is a subgroup of G and A is a Z-set. If H and K
are subgroups of G with H
Hs H H K KS H KA, where HS, K are subgroups of S,
A
S
and HA, K are Z-sets such that H
A, then HflK
(HSfl KS H (HAfiKA).
A
A A,
=
KA
LEMMA 2.
Let G A H B be a Z-factorlzation of G.
H then H A H (H fi B).
that A
If H is a subgroup of G such
(0)
(I) S (I) H A (I)
(n) S (n) H A (n) D
S H A m G
LEMMA 3. Let G G
D...DG
(2.1)
(n+l)
(n)
<0> be a series for G with S(0) S(1)D
G
S
<0> subgroups of G and
A A (0) m A (I)
A (n)
{0} Z-sets There exists a refinement of (21) which is a
...
composition series for G and the subgroups in this refinement have the same properties
as
(21),
H, in the refinement has the form H
i.e. any subgroup,
H s @ HA where HS
K implies H
Ks and
S
A.
is a subgroup of S and H
A is a Z-set, HA K A, and H
Furthermore, if H c K are successive groups in the refinement
HA!K
then either
HS= KS or
KA-
HA
(i+l)
c G with G i)D
It suffices to show that if there exists
G
(i)
(i+l)
H with S
A (i) D D A (+I)
D_
a Z-set and either
D_ S
PROOF.
then
A (i)
or
0 < i < n.
(i)
(i) @ (i)
(i+l)
S
A
Case I. Suppose that A (i)
A (i+1).
(i)
we have
A
H (6 0 S
by Lemma 2.
this case we have
A (i)
Case 2. Suppose that
A (i+l).
(i)
(i) A (i+l)
S
S
without altering the
(i)
(i)
(i)
G
S
6 S (i) H A (i+1)
A
Consider S
S
S
=
(i)
be such that G
Let fi
then by Case 1
(i) H (
S
G
(i+1).
has the required form.
N
A(i)).
Clearly A
(i)
6
(i+l)
6
(i)
0 < i
such that C
m
0 S
(i)
n
(i)
S
=
(i+l).
We can insert the subgroup G
C
(i+1)
In
(i+l)
+
structure of the series, i.e. we have
C
(i+l)
S
(i+i)
H A (i+l).
(i+l)
G
If fi m
(i)
D
D
then by Lemma 2,
Finally, if G
(i) D (i+l).
A
A
In this case we have
fi
If
we are done.
This completes the proof.
THEOREM I.
and if G
fi
H A
Clearly S
i(i)
=
(i+l)
Then for any
(i))
s (i)
S
[5] If G
N(1)H...H N (r),
(IN(i)I,IN(J)I)=I for
B
(I)
H...H B
(k),
where each B
(j) is
where each N
(i) is a
Z-set,
a subgroup of G,
is j, then
(a)
B (i)
(N (1) n B (i)) H...H (N (r) n
B(i)),
(b)
N (j)
(N (j) 0 B (I)
B(k)),
! i ! k,
and
H...H (N (j) 0
j
r.
< j
< i <
k,
r, such that
(i)
FACTORIZATIONS OF FINITE GROUPS
269
The following lemma is a direct consequence of the Second Isomorphism Theorem.
LEMMA 4.
Let U,
and K be subgroups of G with
Ul,
u 1].
[u:
Let S be a subgroup of G.
Then [U N K:
UlC U.
<-
UI0 K]
We will say that S is homogeneous if S is a direct sum
of cyclic groups of the same order.
Theorem 2, which is a generalization of the following result of L. Fuchs [4], p.79),
can be readily verified.
(Fuchs) Let S be a pure homogeneous subgroup of G of exponent
pkG _c
subgroup of G satisfying
H and S N H
pk
and let H be a
If M is a subgroup of G maximal with
<0>.
S OM.
<0> then G
respect to the properties H M and M 0S
n
@ S. be a pure subgroup of G with Si,
THEOREM 2. Let S
i & n, homogeneous
i=l i
of exponent
kl>k2..>kn, and let U G. There exists a subgroup, T, of G with
pki,
U
T and G
c
[pkjG +
S @ T if and only if
>
S.i + U]
S.
j & n.
<0>,
j
REDUCTION TO THE CAUSE OF P-GROUPS
3.
Consider the series
G
G
(0)
S
where S
A
S
(0)
S
(I)
(i),
(I)
...
S
S
(I)
A
(n)
(I)
D...m
G(n)=s (n) O
A
(n)
<0> are subgroups of G and
(0)_ T(1)_ ..._ T(n)_
such that T=T
<0> and
(i)
S
T(1)
(i) eT (i)
-..2
AffiA(0)A(1)...RA (n)
G(i)=s (i)
Let us note that by Proposition 1 [3] there exist subgroups T
G
(3.1)
m <0>
We say the series (3 I) admits replacements if there exist sub-
{0} are Z-sets.
groups, T
G
m
0 < i < n
T(n)
T
(i),
0.< i <.n.
(i) such that
However it is not necessarily the case that T
(0)
D
This problem will be treated in the next section
A group C admits replacement if every series for G of the form (3.1) admits
replacements.
The following theorem enables us to restrict our investigations in this
area to the case of p-groups.
THEOREM 3.
Let G
where the G
G
P
P
P
are the primary components of G.
G admits
replacement if and only if for each p, G admits replacement.
P
G for some p and let
PROOF. Suppose G admits replacement. Let H
P
(0)
H (0)= S
H
A (0)
be a series for H with S
A (0) D_ A
O A
D_
H
H (m)= S
Then G
G
(m) @ A (m)
s (1)
(0)
(I) _m..._m A (m)
(0).
(1)
(0)
D_ <0>
Consequently the series H
S
(I)
(I)
...
A
S
...
(m)
Define K
(S (0)
A
(0)
m
(m)
S
(m)
A
(m)
=
<0>
<0> subgroups of H and
{0} Z-sets.
K)
H
H
, pGp,
P
(0)
S
so that G
(0) @
K
H
A(0) m H(1)=
S
=(S(0)@ K)
(I)
A
(I)
is a series for G which by hypothesis admits replacements.
H
(0)
H
(I) m...m (m)
H
m <0> admits
replacements.
270
E.E. OBAID
Conversely, suppose G
G
G
(0)
S
(0)
A
(0)
:>
,
G
(i)
P
(0)
p
...
"I’(%
S
(i))
S
D
(i)
P
(I)
p
S
S
S
(n) @ A (n)
n <0>
"n’(%
Alt
G’;’{{%
P
(n)
S
D...D
D
p
p
A
(i)
<0> and A
P
(i)
P
A (i)
p
G (1)= S
(0)
A
P
is a series for G
"-
G(i)),
(i)
(0)
P
... G(n)=
(I) @ (I)
A
<0> subgroups of G and A0(
(i)
D {0} Z-sets.
For each p define G
to be the p-primary component of
i)"
i)"
i}-"
i)-"
<
<
$ G
0
G 0 G
i
n, so that G
and G
By Theorem we have that
P
P
p P
(S (i) n G
(A (i) a
n.
0
i
P
P
G (0)-- S
P
P
G
S
Let
(n)
A
Define S
S
(I)
"-
i}"
G
G
S,0.(1
be a series for G with
D...D
admits replacement for each p.
P
(I)
...
...
(I)
Ap
(0)
(I)
A
P
0 & i & n.
G
p
P
_
".
(n)
Ap
G (n)= S
P
Clearly
{0}.
Thus for each p,
(n)
A
P
which by assumption admits replacements.
(n)
P
<0>
Thus for each p
there exist subgroups T ti) such that
P
(i)
T
Z
P
(0)
Z
P
(1)
p
...
Z
(n)
<0>
p
and
(ii) G (i)= S (1)
P
P
Define
T,i,(
G
(1)
(i)
G
P P
e
0 & i & n.
T,i.(
Z
P
T (0)
that T
D
T (i)
P
0
P
T (I)
O (S
P
0 & i < n
.<
i
T(n)
D ...D
(i)
P
<.
T
D
From (i) we have
Note that this sum is direct.
n.
<0> and (ii) implies that
(i))
P
S
P
(i))
P
e
T
P
(1))
P
S
(i)
T
(i)
This completes the proof.
EXTENDABILITY
S’ e A’
Let G S A D G’
S’ e T’, where S’ _c S are subgroups of F, A’ _c A
are Z-sets, and T’ is a subgroup of G’. We say T’ is extendable to G if there exists
T, a subgroup of G, such that T’
T and G S T.
4.
The following theorem provides a necessary and sufficient condition for extenda-
T’ when G is a p-group.
Let G be a finite abelian p-group of exponent
and let G’ be a
S @ A, G’
S’ @ A’
S’ @ T’ with S’
subgroup of G. Suppose G
S, subgroups of
G, A’
A, Z-sets, and T’ a subgroup of G’. T’ is extendable to G if and only if
bility of a subgroup
pk
THEOREM 4.
there exist subgroups,
S’
0
piS)
T i, 1
i
Ti,
such that
k-l.
T’
T1
T
2
...
Tk_ 1
<0> and G’
pG
FACTORIZATIONS OF FINITE GROUPS
PROOF.
By Lemma
Assume there exists subgroups,
piG
G’
pi,
of exponent
with T
plS)
(S’
G’
fl p iG
Ti;
with
we have that
@ T i, I & i & k-l.
i
k.
S
T we must verify
S.1
i<k
and
(p
JG +
T’
fl
T
plS)
(A’SpiA),
@
...
+ T’)
S
i<k
S. + T’) fl S.
+ T’)
i
.<
<0>,
k-l.
k
@ S
where each S is homogeneous
i
i=l i’
Let S
S
k
.<
<- i
<0> and
Tk_
By Theorem 2, to prove the existence of
1
T’ and G
k
(p G +
(S’
271
<-
j
S
(4.1)
<0>
k
T, of G
a subgroup,
k-I
(4.2)
i<j
For (4.1), suppose s
k
s
i<k
T’.
G’ we have t’
a’
0 by the definition of S
s
k
i<k
+
s..m
+ t’,
s’ + a’, s’
s
i
S
e
A’.
S’, a’
Thus s
But then t’
A.
T’
k, t’
< i
i’
X
k
i<k
S’ S T’
However, since S is a direct sum of S l,
+ k.
Let I & j
0
i
s.x + s’
<0>.
$2,...
Since
+ a’
Consequently,
S k, we have s
Together these imply that s
0. Hence (4.1) is true.
k
Note that
<0> if i
j. Thus we have piG
s
pJs.
< k.
so that
i>j
k
0
+
0
+
pJs i pJA.
Suppose
p3g +
s.
3
where s
we have
t
Thus
s.
3
"
3
G’
piG
where
t’
i>j
p
S
T’.
+
s.
+ t’
G’
S’
1
i<j
Since
T’
A’
$
(4.3)
A
a
+ s’ +
s.
pJ a
+
pOa +
a’.
Therefore
pJA
- pJs.1 +
n A’ c_ G’
fl
pJG
(4.4)
+ s’
s.
(4.5)
i>j
pJA
(4.4) that a’
N A’. By hypothesis,
Therefore (4.4) implies that a’
+tj’
is clearly a Z-set we have from
pJs)
P3i
i>j
A, t’
s
-a’
(S’ fl
s’ +
i>j
i<j
i>j
A’
-
a
p3s. +
i>j
s.
pJA
+
S
pJa
Since
I
i & k, a
1
Si,
i
pj s i
+ t’
s.
i<j
S’
jew
T. with T.
O
3
pj S
si + tj,
c
S’
where s’
T’.
i>j
i>j
pJs i
+
i>j
and t
pJ i
j
T
T’
j
S’ and t.
j
PJi
But then (4 3) becomes
S’ and t.
j
T’
Consequently
272
E.E. OBAID
pJs I
0 by the definition of S’ Q
T’
and we have
s’
i>j
Substituting this expression for s’ into (4.5) we obtain
pJ s.
E
s.
3
+ E
i
i>j
E
s.
i>j
i>j
pJ s.I
so that
sj
--0.
T’
Conversely, suppose there exists a subgroup
pG
Lemma I, G’ N
2
(S’0 p s @ (T’
pk-iT
T’0
p T
i
0
piT)
1
Hence (4.2) is true.
Clearly T’
k-l.
i
S Q T.
such that G
D
Thus we can complete the proof by choosing T
1
By
T’O pT DT’ N
T’ N
pT
li&k-l.
A, where S is a subgroup of G and A is a Z-set, is
G’ is a subgroup of G such that G’
S’ @ A’
S’
T’, where
_c S, T’ c_ G’, and A’ is a Z-set contained in A, then T’ is always
Let us note that if G
S
an elementary abelian p-group, and
S’
extendable to G.
LEMMA 5.
S’ @ A be a series for G with S’
S subgroups of
G’
that
If T is a subgroup of
S’ @ T then G S T.
G’
such
be a set of coset representatives for S modulo S’.
Q S’ e A
S’ e T S T.
Let
PROOF.
5.
A
S
Then G
G’
S @ A
Let G
G and A a Z-set.
SOME GROUPS WHICH ADMIT REPLACEMENT.
We noted in Section 4 that given the series
G
G
S(0)QA (0) DG (I)= S(1)e A (I) D...DG (n)= s(n) A (n)
(0)
where S
_D {0}
0 <- i
S
(0)
D_ S
Z-sets,
are
(1)
_..._
S
(n)
_D <0> are subgroups and
one can always find subgroups
A
A(0)_A(1)_=..._ A (n)
T (i) such that G (i)ffi
n, although it need not be the case that T
(5.1)
<0>
s(i)
T(0) D_ T(1)D_..._D T(n)_D
T
(i)
<0>.
However, by applying the extendability criterion of Theorem 4, we can ensure that for
(i)
each i, 0 <- i <. n, our choice of the subgroup T
will be extendable to each G
(0)T (n).
<- i, and consequently we will have T T D_ T
(=),"
(1)_...D_
=
We will briefly illustrate how successive applications of Theorem 4 when G is a
p3
finite abelian p-group of exponent
results in Figure 1 since this lattice-type
structure clarifies the proof of the major theorem in this section. By Lemma 3 we may
(5.1) is a composition series for G.
We introduce the following notation to simplify the discussion:
assume that
G! i)
S
(i)
j,
A! i)
(i)
pJG ()
S
(i) N
pJs ()
A
(i) 0
pJA ()
G
FACTORIZATIONS OF FINITE GROUPS
<
where 0 < i < n, 0<
0
<.
<.
i <- n, 0 <-
By Lemma
1,2.
n, j
G!3,i such that
P
hA(i)
’I,
(i) @
S
h.(i)
h.(i)
@
p
3,
w p
ml,
3,
3,
I, 2.
n, j
s!i A!i
3,
G:i( S! i) A!3 i),
we have
3,
G!i)
3,
273
T!i,
j,
3,
and T
(i , will denote any subgroup of phGi such that
h,
h(i) w _(i)
ml, Zh,l, u
p
AI,
< n, u < m < I, 3
1
z, z, h a non-
negative integer.
1,0
,,,,,
,
,,.-
/9,.-z
IGI.-o)
Iln-l)
,.-
6 (")
IA
Gl’zd
I/
, I! !,
"!’1
s,.-]l
IAG(.-I)I
I/
l(n-.I
I/
l/GIn-2) l/Gin-z)
A
i/Gin-I)
/
I,O
Consider a subgroup T
In) such that
By Theorem 4, T
In) will be extendable to G (i), 0
subgroups
such that T
Tj,l
In)
we have G
)=
i
S
In)
j,i’
0
i
We have the following array for the containments of the subgroups
G
pG(n) m_ G In)
l,n-I
In)
l,n
m
G
l,n-2 2..
.:3
_In)
ul, 1
_In)
(n)
Gin)
c.
p2G(n) m_ Gin)
2,n-i m---2,n-2--’’c-u2,1
2,n
In)
Thus if we can find subgroups T
j,i
In)
T l,n
Ul
c
In)
T l,n-I
c
T
Ul
In)
T
T(n)n
2,
2,n-i
c
T
Ul
Ul
Ul
In)
_In)
2,n-2
c
TI,I
’’’ T2,1
s(n) T In).
D
G
TI,0
2,0
1,2.
n-l, j
G(n
j,i’
0
i
In)
Ul
m
G
0 < i < n, j
_In)
l,n-2
_In)
Ul, 0
In)
2,0
.....
_In)
In)
D_
Ul
Ul
Ul
Ul
In)
In)=
< i < n-l, if an only if there exist
_In) with G
2,i
In)
Tl,i
D
(5.1)
in the series
1,2
T
In)
T
In)
such that
n, j=l,2:
274
and
E.E. OBAID
S(3
G(I
3 i
T(3
i
(n) extendable to G (i)
T
i < n
0
i’
G(n)= s(n)O
2
j
T
(n)
we would have
0 < i < n-I
(n-l)
_(n)
Later it will become apparent that we need TI, i extendable to
i
(n)
0& i & n-l. We will show how this can be incorporated in our discussion on T
GI,
Using Lemma 4 with U
[G(nl) G(n]
I,
Note that
U
(n)
G
K
(n-l) has
exponent at most
p2
PGl,i(n-1) PSl,i(n-1) @ T(n-l)l,l,i, 0 < i < n-l.
(n-l) n p2G(i) c G (n) p2G(i)
_(n)
pG
u2, i 0
G
(n)
S
l,n
G
(n)
2,n
(n)
l,n-I
G
c
(n)
G(;I
(n-l)
i’
c
G(n)
l,n-2
c
G
(n)
2,n-I
(n)
2,n-2
Ul
..(n-1
c
Thus if we can find subgroups
T
_(n)
II,i
0 < i < n-i
is extendable to G
such that
(n-l)
n-I
(n-l)
PGI,i
Pl,i
_(n)
Tl,i--
(n-l)
l,i
T(n-
i,i i
p(G n-l) A pG i)
we have the following array
1 2:
j
c
u2,1
_(n)
Ul
(n-1)
c
c--PGI,1
I)("lj, i T-,
_9
.(n-1
PI,O
0<i< n, 0< i’<n-l, j= 1,2,
(n) such
that
.(n)
T(n)
T(n)
Xl,n-2
l,n
l,n-I
U1
(n)
T 2,n
U1
c
(n)
T 2,n-I
G!3,in)
_(n)
(n)
1 2 G
and T
(n)
S
c
2,n-2
(n)
"
(n)
extendable to G
(i),
_(n)
U1
U1
(n)
_(n)
(n-l)
U1
1,1,0
T
(n)
we would have T,
0
i
n-l.
(n)
T2,0
1,1
PSI,i’
T
TI,0
U1
l,l,n-2-""
T
(n)
T I,I
T 2,1
U1
T(n)
(n-)
j,i’ PGI,i
bj,i
c
.(n)
1,1,n-1
with
c
Ul
U1
j
_(n)
c
Ul
.(n-1
c_ P’l,n-2
_(n-l)
_(n)
G(n)
Ul,0 G(n)
I,I
c
Ul
PUl,n-
i <
n-l, we have
i
PGl, i
Since
0 < i < n-I
PUl,i
fl
0
(n-)i
TI,I
if and only if there exists a subgroup
for the subgroups G
(i)
By Theorem 4
and
c
pG
(n-I) c_ _(n) so that _(n)
Ul, i
Ul, i
GI, i
i < n-i
0
(n-l)
PGI, i
Thus
p"
[ I,
G
,i
I,i
0 < i < n
0
i’
n-l,
(
n-l)
extendable to G,
i
,i
0 < i
n-1
In particular, we would know there exists
FACTORIZATIONS OF FINITE GROUPS
a subgroup
T
(n-l) with T (n-l)
T
(n) and G (n-l)
(n-l)
S
275
T (n-l).
$
However,
(n-l) is extendable to G (i) 0 &
we must ensure that our choice for T
n-2
i
n-l)
and then to T i), 0 < i < n-3, we obtain
Appyling the previous argument to T
Figure I. We remark that lattice-type structures similar to Figure I can be obtained
pk,
for finite abelian p-groups of exponent
Such
where k is any non-negative integer.
3
structures become rather complicated when the exponent of the group exceeds p
The following definitions will facilitate references to Figure I.
DEFINITION 1
The row for G
DEFINITION 2.
The row for
(i)
m G_
2,0-
m
2,1-
_(i)
DEFINITION 3
pG
,0
m
pG ,2
()
2,0’
and
0 & i & n, is the series
,.(i)
0 & i
u2, 0
(i)
m
pG i),0’ 0 .<
i
.<
n-l, is the series
P’l,i
(i),
The sub-figure for G
pS ,0
n, is the series
2,i
The row for
DEFINITION 4.
G()
G
m
u2,2-D
(i)
<. n,
i <-
i-I
<.
0 < i < n, consists of the rows for
’
n-l.
We say the sub-figure for G (i)
DEFINITION 5.
S
(i)
complete if
(i) for every subgroup H in the sub-figure for G (i)
S
a subgroup of S
H
H
S
H
(i),
a Z-set,
AH _c A
(i),
$
A (i), 0 & i & n,
H=
is
SHgAH,
there exists a subgroup T
H such that
TH,
$
(ii) for all sub-groups H, K in the sub-figure for G (i) with H
c
K we have TH c_ TK.
Let us note that, by the construction of Figure I, if the sub-figure for G
1
=<
n, is complete then T
i
G,
i-l)
0-<
-<
T
T
(i)
i)
,’ 0 <=
T(i)m
is extendable to G
(i-1)
to
i"
DEFINITION 6.
(i)
(i)
(1),
The row for G
(i)
& i
n, is complete if there exist subgroups
-< i, such that
T
i)
m_ T ,1
,0
i) m_
i)
D__m T ,i P- <0>,
, ,e
S
(iii) T (i) is extendable to
G.i-l.(l
T
(
and
0
,
T; i)
.<
.<
i,
is extendable to
Gi I)
0
i
276
E.E. OBAID
We will say that the sub-figure (row) for G
(i)
can be completed if we can prove
(T (i) _(i)
Ii, discussed in the definition for "The subH
(i)
is complete."
figure (row) for G
PROPOSITION i. Let G be a finite abelan p-group and let
the existance of the subgroups T
G
G
(0)
A
S
D
G
series for G with S
A
A
(0)
A
D_
(I)
<" p
[Gi .(i+l)]
-I,
<,.>
ro<-,:
< p’ 0 <
,
<
-2,
-
Let G (’i)
Then
<gl >
_(i)
@
o<-,+,>]
e
<Pgk>
k=1
and
o<,..<
I,
[G(i):
i
<pgj>
r
i)
pG
Ul, i
<g2 >
@
i+l
Since
<gk>.
k=l
G(i+l)
following
n:
(b)
2,
{0}
D_
Gl,i+l
,i
PROOF.
S
D_
D_
<. i’ .<
0 <- i <- n-l, 0
_
(I) O A (I) D
(n) S (n) e (n) D <0> be a
D G
A
composition
(n)
(0)
(I)
_D S
a S
S
<0> subgroups of G and
(n)
Z-sets. The
A
statements are true for
(I)
pG
bl,i+1
}
p
we have
=<
for some j,
<gk >
(i+l)
(i+)
G
i+,
i+l)
<pg 1 >
j
=<
<Pg2 >
r.
e...i
<p2gje...e
<Pgk >
If
o(gj)
p
This proves
then G
i)
G
,i
If
,i+l
o(gj)
> p
then
G
i)
,i:
G
i+l)
,i+l
p"
(a).
Each of properties (b) through (f) can be deduced from Lemma 4 by choosing U, U I,
and K appropriately as follows:
(b)
U
G
(i)
,,
G
U1
U
1
(i+l)
G
,,
K
G?
0 <
,’+i’
< i+l
FACTORIZATIONS OF FINITE GROUPS
(d)
(,.(t
U---I,
(e)
U
G(i
UI
I,
(f)
U
Gil
U
G
1
G
G
U1
Observe
that[G (i) G (i+l)]
Thus
i),i: -l,ic(i+l)
[G
i+l)
,i
(i+2)]
c
-2,i
G
S @ A
<
n-l, [
.<
i
<. n-l,
D
.<
0
0 <
G
2,i:
-< i.
< i+l
(i)
c
G
(i+l) and
p2G(i)
i+l)
c_ pG
c_ G (i+2)
0 -< i <-n-l, and
,i
n-2
S
(0)
S
(I)
A
(I)
S
m_
(I)
D...D G
m_..._m
(n)
S
A (0) _m A (I) _m...m_ A (n) D_ {0} Z-sets.
A
0 -<
Let G be a finite abel:an p-group and let
(I)
G
series for G with S
0
G(
K
p implies that pG
0 < i
PROPOSITION 2.
(0)
G
i+l)
I, 0 <. i
I
G(i+l)
i),’+i’ K
.< i’
0 <-
K--G
,+
277
.<
i, 0
.<
’ .<
(n)
S
(n) O A (n)
D
<0> be a composition
<0> subgroups of G and
D
The following statements are true for
i-l.
,and
(b)
If
(i)
rG[ 1,’+I
G
(i)
,t
We have
PROOF.
1
i+1)
rG(i-,l+) 1
[ I,
,’+I]
)]
p
,.(i)
t,
_c G
2,
[(i-l)
then
,.(i) -c
"1., +1
i)+1
G i),’+I]
1 and
_(i)
G(i+l)
Therefore
1 and
,’+I: i,’+i}
2,
,,:
G
,
p and
+I
Proposition 1 we have that
G
i+l):
,’
I, ’+lJ
[ I,
G
i+l)
]
,’+lJ
p"
I.
G
S
,,:
i)
G
(i D G (i+l)
I,’
By (d) and (e) of
Consequently
I,
’
I.
We can eliminate from consideration several combinations of indices in Figure
since by Proposition 2 it is impossible for them to occur.
LEMMA 6. Let W, X, Y and Z be subgroups with the following properties:
W O Y
(i) W X, Y X, Z
(ii) [W:Z]
[X:Y]
p
(iii) X
S
x Ax, W Sw
.
Aw
SW, SX, TW, Ty are
k
with A E AX, hy
W
x.
where
(iv)
Z
S
w
@ T
Z
with T
Z
S
w
, TW, Y
S
x
subgroups with Sw
Tw and T Z
Ty.
@
c
Ay
S
S
and
x
X
, Ty,
AW, Ay, Ax
are Z-sets
278
E.E. OBAID
Then X
S
W +
TX, where TX
X
we have Z
By Lemma
PROOF.
Ty.
T
(S
W 0 Y
0 S
W
X)
O (A N
W
Ay)
Ay).
@ (A
S
W
W
The following diagram illustrates the relations between the subgroups W, X, Y, and Z.
SW @ AW
W
S
W
Y
TW
*
Z
We will first show that X
W+Y
[W:Z]
p we must have X
W
S
W N Y
S
* Ay
X
S
@ (A
W S
W
+ Y. We have Y _c W+Y c_ X
S + (Tw + Ty).
x
We will complete the proof by showing that S & (T +
w
x
* Ty
X
Ay)
S
T.
W
and [W:Y]
Ty)
p.
Since
<0>.
Let
s
w + ty, sx
We can write tw
ay
w
Ay
Aw 0
Similarly,
_
s
THEOREM 5.
c__ Z.
Consequently we can write aw
Since
(0)
A
S
A
A (I)
_..._D
D
S
be a series for G with S
(0)
A
s
Z so that X
t
x
_c Tw
Tz
and T
S
w
s
A (n)
G
(I)
(0)
D_ S
S
(I)
(I)
(I)
A
D_..._D S
_D {0} Z-sets.
If
p2A
D_ <0>
AW, ay
s
and
W
e
S
x
(n)
+ sw
s
W
pk, k
s(n)@
W
t
w + sw
s
W
(n)
T Z.
Z
s
=> I,
A
Ay.
e
<0> we must have s
w
D...D G
(n)
e
w + tz, sw
Hence s
0.
W
s
Let G be a finite abelian p-group of exponent
O
S
s
s
s
ty
(5.2)
tye Ty
Tw,
e
+ ay, sW e SW,
Sx, aW
w + aw, ty
s
x sW + aw + sx + ay and we have sx s X
w + sw + t Z.
But then tw
e
w
s
Thus (5.2) becomes
-a
Sx, t
t
X
<-
n, 0
A!j,i)
=<
<.
i, j
>. 2,
h
pJA () S p2A
A (i) n
0
x
and let
D
(5.3)
<0>
subgroups of G and
{0} then the series (5.3) admits
replacements.
PROOF. By Lemma 3 we can assume that (5.3) is a composition series for G.
that for 0 <- i
Note
>. I,
{0},
and
p
h.(i)
A1,
ph+lA() p2A
_(i)
G!i)
bj,
j,
Thus we have
and
Consequently we must have
T!
i)
3,
{0}.
phGi I phsi)0
,, "
(i)
h,l,
i
n, 0
"
<0>, 0 < i & n, 0 <
i
j > 2
h
i
j > 2
h >
We use a "backward induction" on i to show that the sub-figure for each G
0
i
"
(i),
n, can be completed.
For i
IG(n)l
(n)
n,
figure for G
p implies that G
is trivially complete.
(n)
0.
is cyclic of forder p.
Thus the sub-
FACTORIZATIONS OF FINITE GROUPS
Now assume the sub-figure for G (i+l) is complete
T!i3, Th,’(i)
l,
comments on the subgroups
the row for G
(i)
0 <
In view of the preceeding
i, j > 2, h > i, if we can complete
Ti+l)
c, Ti,
in such a way that
279
<- i, T (i+l) c_ T
0 .<
(i),
(i)
then the sub-figure for G
will be complete as well
(i+l)
By Lemma 3 we must consider two cases (I) A (i)
S
A (i+l) and (2) S (i)
Case (I). By hypothesis the sub-figure for G (i+l) is complete so that there exist
(i+l)
i+l) D
T
subgroups T
such that
D. D T
D_ T
,I
,0
,i+l
G
(i+l)
S
extendable to G
K
pG
()
(i+l)
(i)
T
(i+l)
_(i+l)
< i
Gi), Si @ _(i+l)
i-I,
i+l)
0 <
< i
T,
i+l)
T
G
0 <
(i+l),
Ti
< i
.p,
[G ,: (i+l)]
Ul,
If
c.(i+l)]
[Gi),: -I,
Ti
_(i+l)
i-
I,
0
0 <
I
< i
< i
0 <
G
(i+l) is
(i)
< i
Applying
this time with G
G
(i)
Thus we can complete the sub-
T
(i)
T
(i+l)
by (b) of Proposition I.
we can complete the sub-figure for G (i) by choosing
-< i and extending T (i+l) to G (i)
.<
Setting H
and T
< i, we have
T (i+l).
<
< i+l
A (i+l)
Ai
Again applying Lemma 5
(i)
(i)
T T
we have G
S
(i)
T(i+l) 0
figure for C
by choosing
-i,
i)
<
Case (2)
We have
0 <
G’
0 <
<- i.
0
we conclude that
G,
G’
Ul,,
(i+l),
I
i+l)
Gi
is extendable to
in Lemma
_(i)
Lemma 5 with G
i+l)
-I,
i-l,
0 <
S, T,
c(i+l)
(T i+l)
is extendable to G
(i)
by the hypothesis that the sub-figure for G (i+l) is complete).
Now suppose there exists
such that
o
,
o<
for
for 0
This situation is illustrated in Figure 2,
subgroups
2
G!3,
r) and
G,
T!3,
r)
since
<.
-< i
-<
o
where for simplicity
(r)
T
<0> 0
h,l,
n 1
we have omitted the
< h < k-2
j & k-l, r
i+l, i, i-l. The numbers 1 and p in Figure 2 represent indices
i+l)j
i)
< i. As remarked previously the hypothesis
We can choose =j
<
for
o
T, T,
that the sub-figure for G (i+l) is complete implies that
T ’o
i+l) can be extended to
This extension is indicated in Figure 2 by a single arrow
Setting X
Z
W N Y
G
I,
,+i’
0 <
Y
Ul,
<.
o
-i
w
i, +I’
so that
0
o -i,
G’
we have
W,X,Y,Z satisfy the conditions of Lemma 6
i)
o
E.E. OBAID
250
, ,
i)
Thus we can apply Lemma 6 to obtain G
<
0 <
Z
Gi+l),0
W A Y
taking X
We can again apply Lemma 6
-I.
o
to obtain G
(i)
S
T ,’ where
S
(i+l)
T
(i),
G
where T
These sums are indicated in igure 2 by double arrows.
and
Ti+l) Til
c
0 <
< i
(i)
(i)
+1
(i+l)
+T l,i
G
T i),0 + T(i+l)"
i)
,0’ Y
W
(i)
(i)
Clearly,
Thus the sub-figure for G
_
T
I,
G
(i+l)
..
.(i)
I,0-
is complete.
If G is a finite abelian p-group of exponent less than or equal to p
COROLLARY I.
2
then G admits replacement.
Let G
PROOF.
G
(0)
be a series for G with S
A
A (0)-
A
(I)
2...
A
(n)
A
S
(0)
S
D
G(1)=
(I)
S
R {0} Z-sets.
A(1)D...D G(n)=
(I)
...
S
(n)
S
S
(n)
A
(n)
D
<0>
(5.4)
<0> subgroups of G and
p2A
We have
By Theorem 5 the series (5.4) admits replacements.
{0} since by hypothesis
p2G
<0>.
Hence G admits replacement.
RELATION TO A VARIATION OF A METHOD OF A.D. SANDS.
Our terminology will be the same as in [3] when referring to factorizatlons which
are obtained by the variation of Sands’ method.
The following Proposition can be readily verified.
D <0> be a series for G with coset
PROPOSITION 3. Let G K 1D K2 D...=
is a subgroup (Z-set) then
Hn
If H
n, K
representatives H i I & i
n
i
6.
Kn
Hi+2
is also a subgroup
Hi+2 Hi+4
THEOREM 6.
(Z-set).
Let G be a finite abelian group which admits replacement and let
D <0>
(6.1)
K 1D K2 D...D
be a series for G. If G
A B is a Z-factorization of G arising from (6.1) then there
S @ T arises from the series (6.1).
exist subgroups S, T such that the factorization G
PROOF. We will assume that n is odd since the proof for n even is similar. We will
Kn
G
proceed by induction on the order of the group G.
If
IGl=p,
then G
G
<0> is the only Z-factorization of G.
which this factorization arises we must have G
K1
S, T
K.1
Thus in any series from
<0>, i Z 2.
FACTORIZATIONS OF FINITE GROUPS
281
Assume the theorem is true for groups of order less than G. Let G A @ B be a
By Lemma 5 of [3] we may assume
Z-factorization of G arising from (6.1).
A
H @ H3
B
H 2 @ H 4 @ H 6 e...@
Hi
where 0
H 5 @... Hn,
i
n.
3
+ H 5 +...+
(H 3
H 5 e...@
We have the H
Hn_ I,
Hn is
a Z-set by Proposition 3.
Hn
(H2 e H4
Thus
K2
is a Z-factorization of
K
IK21 IGI
K 2 arising from the series
K 3 m...m Kn
2
(6.2)
m <0>
implies, by the induction hypothesis, that there exist subgroups, S’,
<
S’ @ T’ arises from the
such that the factorization K
2
Lemma 5 [3] there exist transversals H
T’
H3 @ H5
H’.I’ 2
where 0
n
K i+l
i’
<.
series
(6.2).
Thus, by
n, such that
n-l’
H’n’
and
Hi
(6.3)
2 <i<n-I
Ki+l
HI are coset representatives for Ki modulo Ki+ I,
and
Hi
H’n
H
Hi
Ki
.
@
2 -< i
T’
< i < n.
Note that K
n
since both
’’’ Hn_l
(6.3) successively, starting with
H’n-I
Hn’ H2’ H4’ H6’
2 < i < n.
n-l, we see that we can choose H I
i
as coset representatives for the series
H3,
Using
H5
(6.1) to obtain the
factorization
(H
G
H
e...e
H3
Hn
(H
By Proposition 3, H i
Hi+2 e...@
n-l.
is a subgroup, i
2,4,
s(i)
S
A
A
=H’.
I
"n-l(
(i)
"n’(
Hi+2
e...e H’n-l’
H
Hn
4’ e...e H’n_I)
is a
Z-set, i
Set
i
2,4
n-3
H’n-I
Hi @
Hi+2
Hn
Kn
...e Hn
i
1,3
n-2
A
S’
1,3, ...n, and
(6.4)
H!I Hi+2’ e...e
E.E. OBAID
282
Then the series (6.1) can be written as
G=KI--S(2)
where S
A
(I)
D
K2--S(2) A(3)D K3=S(4)
S’ and A (I)
(2)
S
S
In general,
A.
(i)
$
,.. Kn- Iffis(n-I)A(n)DKnA(n)D <0>
A (3)
A (i+l) for i even, 2
(i+l)
A
(i)
.<
for i odd,
By hypothesis G admits replacement.
<-
.< n-I
i
Kn
<-n-2,
i
A
(n)
Thus there exist subgroups T
(I)
D_
T(2)D...D_ T (n)
such that
S
S
Define
H"n
K
so that
S
@ T
(i+l)
A (n).
(n)
T
(i)
(i)
@ T
(i) for
i odd,
<.
IT(n)l
i -< n-2, K
n
(n)
T
A
Hn
(n)
We have
(n-l) @ (n-l)= S (n-l) @ (n)
T
A
IT (n-1)l
.< n-l,
for i even, 2 .< i
But T
(n)
c
T
S
(n-I).
(n-l)
@ T
Therefore T
(n)
(n-I)
T
(n)
Next,
Kn-2
S
(n-l)
T
(n-2)
T (n-l)
c
T
(n-2)
a set of coset representatives for T
If we choose H"
n-2
T (n-2)
H"n-2
H"n-2
@
@ T
(n-l)
Kn_ I"
Thus
Kn_ 3
S
H"n-2
H"n-2
(n-3)
T
(n)
H"n-2 + H"n’ and Kn -2
Kn-3
S
c
T
(n-3).
T
(n-3)
S
(n-3)
(n-3) @ T (n-2)
H’n-3
(i)
K
n
H"n
(n-l)
H’n-I
<.
(n-2)
(n-2)
T
S
(n-l) @ T (n)
Kn_ 2
S
(n-l)
T
(n-2)
(n)
modulo K
n-I
(n-3) and
H’n-3
(i-l)
Kn-2"
(i) and (i-2)
T
T
i ffi< n, are as follows:
Hn
Kn-I
Kn_ 2
T
IT(n-3)l IA(n-2)l IT(n-2)I.
imply that
T
H"n-2
(n-2) and
for i odd we have T
so that the factorizations of K i,
(n)
T
A
Hence we have that T
In general, given T
S
is also a set of coset representatives for
Kn-2 s(n-I) T(n-2) s(n-l) A(n-2)
But T (n-2)
(n-2) modulo (n-l) we have
T
H’n-I @ H"n
H’n_I H"n_2 e T(n) H’n_I
e
(H"n-2 H"n)
H"
@ T
(i)
FACTORIZATIONS OF FINITE GROUPS
(n-3)
(n-2)
(H’ 3
n-
@ T
Kn- 3 S
(n-3)
(n-4)
T
Kn- 4 =S
K3
K2
K
S
(4)
S(2)
S
(2)
T
T
T
(3)
(3)
(1)
S
(n-3)
(4) @ H @ T <5)
3
(H
S
S
H’_I)O
(Hn- 2 H")
n
n
H"n_4 T(n-2) ffi(H-3
(2)
e
H4
(H
e..e H’n-I +
H
T
(3)
T
(I)
T
H’ @ H H’; e...e H"n
which arises from the series
1.
+
H’n_l)
(H"n_4 O H"n_2 H")n
H e...e H’n_l) + (H
H3 H5
S
(2)
H
5’ O...@ H")n
.e H")
n
e..e H’n-1
(H H4
We can complete the proof by defining S
283
H @
H1 H3
H e H6 @.
2
to obtain the factorization G
S
.
.e H")
n
H’n- and
T, S _c G, T _c G,
(6.1).
REFERENCES
SANDS, A.D., On the factorization of finite abelian groups I, Acta. Math. Acad.
Sci. Hung.
(1957),
65-86.
2.
HAJOS, G, Sur la factorisation des groupes abeliens, Casopis Pest. Math. Fys.
7__4(1950), 157-162.
3.
OBAID, E.E., On a variation of Sands’ method, Internat. J. Math. & Math. Sct.
4.
FUCHS, L., Abelian Groups, Pergamon Press, 1960.
OKUDA, C., The factorization of Abelian groups, doctoral dissertation, the
Pennsylvania State University, University Park, PA, 1975.
9(3)(1986), 597-604.
5.