Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 760246, 17 pages
doi:10.1155/2012/760246
Research Article
The Number of Chains of
Subgroups in the Lattice of Subgroups of
the Dicyclic Group
Ju-Mok Oh,1 Yunjae Kim,2 and Kyung-Won Hwang2
1
Department of Mathematics, Kangnung-Wonju National University,
Kangnung 210-702, Republic of Korea
2
Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea
Correspondence should be addressed to Kyung-Won Hwang, khwang@dau.ac.kr
Received 9 May 2012; Accepted 25 July 2012
Academic Editor: Prasanta K. Panigrahi
Copyright q 2012 Ju-Mok Oh et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We give an explicit formula for the number of chains of subgroups in the lattice of subgroups of
the dicyclic group B4n of order 4n by finding its generating function of multivariables.
1. Introduction
Throughout this paper, all groups are assumed to be finite. The lattice of subgroups of a given
group G is the lattice LG, ≤ where LG is the set of all subgroups of G and the partial
order ≤ is the set inclusion. In this lattice LG, ≤, a chain of subgroups of G is a subset of LG
linearly ordered by set inclusion. A chain of subgroups of G is called G-rooted or rooted if it
contains G. Otherwise, it is called unrooted.
The problem of counting chains of subgroups of a given group G has received attention
by researchers with related to classifying fuzzy subgroups of G under a certain type of
equivalence relation. Some works have been done on the particular families of finite abelian
groups e.g., see 1–4. As a step of this problem toward non-abelian groups, the first author
5 has found an explicit formula for the number of chains of subgroups in the lattice of
subgroups of the dihedral group D2n of order 2n where n is an arbitrary positive integer. As a
continuation of this work, we give an explicit formula for the number of chains of subgroups
in the lattice of subgroups of the dicyclic group B4n of order 4n by finding its generating
function of multivariables where n is an arbitrary integer.
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Discrete Dynamics in Nature and Society
2. Preliminaries
Given a group G, let CG, UG, and RG be the collection of chains of subgroups of G, of
unrooted chains of subgroups of G, and of G-rooted chains of subgroups of G, respectively.
Let CG : |CG|, UG : |UG|, and RG : |RG|.
The following simple observation is useful for enumerating chains of subgroups of a
given group.
Proposition 2.1. Let G be a finite group. Then RG UG 1 and CG RG UG 2RG − 1.
For a fixed positive integer k, we define a function λ as follows:
λ xk , xk−1 , . . . , xj
λxk : 1 − 2xk ,
: λ xk , xk−1 , . . . , xj1 − 1 λ xk , xk−1 , . . . , xj1 xj
2.1
for any j k − 1, k − 2, . . . , 1.
Proposition 2.2 see 5. Let Zn be the cyclic group of order
β
β
β
n p 1 1 p2 2 · · · pk k ,
2.2
where p1 , . . . , pk are distinct prime numbers and β1 , . . . , βk are positive integers. Then the number
β β
β
RZn of rooted chains of subgroups in the lattice of subgroups of Zn is the coefficient of x1 1 x2 2 · · · xkk
of
φβ1 ,...,βk xk , . . . , x1 1
.
λxk , . . . , x1 2.3
Let Z be the set of all integer numbers. Given distinct positive integers i1 , . . . , it , we
define a function
x1 , . . . , xk −→ y1 , . . . , yk ,
2.4
if / ij ∀j 1, . . . , t,
ij for some j such that 1 ≤ j ≤ t.
2.5
πi1 ···it : Zk −→ Zk ,
where
y x ,
x − 1,
Most of our notations are standard and for undefined group theoretical terminologies
we refer the reader to 6, 7. For a general theory of solving a recurrence relation using a
generating function, we refer the reader to 8, 9.
Discrete Dynamics in Nature and Society
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3. The Number of Chains of Subgroups of the Dicyclic Group B4n
Throughout the section, we assume that
β
β
β
n : p1 1 p2 2 · · · pkk ,
3.1
is a positive integer, where p1 , . . . , pk are distinct prime numbers and β1 , . . . , βk are nonnegative integers and the dicyclic group B4n of order 4n is defined by the following presentation:
B4n : a, b | a2n e, b2 an , bab−1 a−1 ,
3.2
where e is the identity element.
By the elementary group theory, the following is wellknown.
Lemma 3.1. The dicyclic group B4n has an index 2 subgroup a, which is isomorphic to Z2n , and
has pi index pi subgroups
api , b, api , ab, . . . , api , api −1 b ,
3.3
which are isomorphic to the dicyclic group B4n/pi of order 4n/pi where i 1, 2, . . . , k.
Lemma 3.2. (1) For any i 1, 2, . . . , k,
api , ar b ∩ api , as b api ∼
Z2n/pi ,
3.4
where 0 ≤ r < s ≤ pi − 1.
(2) For any distinct prime factors pi1 , pi2 , . . . , pit of n,
api1 , ar1 b ∩ api2 , ar2 b ∩ · · · ∩ apit , art b ∼
B4n/pi1 ···pit ,
3.5
where r1 , . . . , rt are nonnegative integers.
Proof. 1 To the contrary suppose that
p
api , ar b ∩ api , as b / a i .
3.6
Then api ur b api vs b for some integers u and v. This implies pi | s − r. Since 0 ≤ r < s ≤ pi − 1,
we have s r, a contradiction.
2 We only give its proof when t 2. The general case can be proved by the inductive
process. Let
K : api1 , ar1 b ∩ api2 , ar2 b.
3.7
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Discrete Dynamics in Nature and Society
Clearly, api1 pi2 ∈ K. Since gcdpi1 , pi2 1, there exist integers u and v such that pi1 u pi2 v 1.
Note that api1 −ur1 −r2 r1 b api1 −ur1 −r2 ar1 b ∈ api1 , ar1 b. On the other hand,
api1 −ur1 −r2 r1 b a−pi1 ur1 −r2 r1 b
api2 vr1 −r2 −r1 −r2 r1 b
since pi1 u pi2 v 1
3.8
api2 vr1 −r2 r2 b ∈ api2 , ar2 b.
Considering the order of K, one can see that K api1 pi2 , api1 −ur1 −r2 r1 b. Since
api1 pi2 4n/pi1 pi2 e,
api1 −ur1 −r2 r1 b
pi1 −ur1 −r2 r1
a
2
b2 an api1 pi2 n/pi1 pi2 ,
−1
b api1 pi2 api1 −ur1 −r2 r1 b
api1 pi2 −1 ,
3.9
we have K ∼
B4n/pi1 pi2 .
By Lemma 3.1, we have
UB4n Ca ∼
Z2n i −1 k p
C api , aj b ∼
B4n/pi .
3.10
i0 j0
Using the inclusion-exclusion principle and Lemma 3.2, one can see that the number UB4n has the following form:
UB4n CZ2n zi1 ,...,it C Z2n/pi1 ···pit
1≤i1 <···<it ≤k,
1≤t≤k
1≤i1 <···<it ≤k,
1≤t≤k
bi1 ,...,it C B4n/pi1 ···pit
3.11
for suitable integers zi1 ,...,it and bi1 ,...,it . In the following, we determine the numbers zi1 ,...,it and
bi1 ,...,it explicitly.
Lemma 3.3. (1) bi1 ,i2 ,... ,it −1t1 pi1 pi2 · · · pit .
(2) zi1 ,i2 ,...,it −1t pi1 pi2 · · · pit .
Proof. 1 Clearly bi1 −111 pi1 pi1 for any i1 1, . . . , k. For any integer t ≥ 2, one can see
by Lemma 3.2 that among intersections of the subgroups of the right-hand side of 3.10, the
group isomorphic to B4n/pi1 pi2 ···pit only appears in t-intersection of the subgroups
api1 , aj1 b , api2 , aj2 b , . . . , apit , ajt b ,
3.12
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p pi2 pit where 0 ≤ jr ≤ pir − 1 and 1 ≤ r ≤ t. Since there are 1i1
· · · 1 pi1 pi2 · · · pit such
1
t1
choices, we have bi1 ,i2 ,...,it −1 pi1 pi2 · · · pit .
2 By Lemma 3.2, one can see that among intersections of the subgroups of the righthand side of 3.10, the group isomorphic to Z2n/pi1 pi2 ···pit only appears one of the following
two forms:
a ∩ api1 , aj1 b ∩ api2 , aj2 b ∩ · · · ∩ apit , ajt b ,
3.13
api1 , aj1 b ∩ api2 , aj2 b ∩ · · · ∩ apit , ajt b ,
where 0 ≤ jr ≤ pir − 1 and 1 ≤ r ≤ t, and each subgroup type in the first form must appear at
least once, and it can appear more than once, while each subgroup type in the second form
must appear at least once, and one of the subgroup types must appear more than once. Let
γ be the number of the groups isomorphic to Z2n/pi1 pi2 ···pit obtained from the first form, and
let δ be the number of the groups isomorphic to Z2n/pi1 pi2 ···pit obtained from the second form.
Then clearly zi1 ,i2 ,··· ,it γ δ. Note that
γ
pi1 ···pit −t
r
j1 ···jt tk, r1
1≤jr ≤pir ,1≤r≤k
k0
t pi
−1t2k
−1tk
t pi
r
r1
k≥0 j1 ···jt tk,
1≤jr ≤pir ,1≤r≤t
jr
jr
3.14
t pir
j1 ···jt
−1
jr
1≤j ≤p ,1≤r≤t
r1
r
ir
t
−1
j1 ···jt
r1 1≤jr ≤pir
pir
−1t .
jr
On the other hand,
δ
pi1 ···pit −t−1
pi1 ···pit −t−1
−1
r
j1 ···jt t1k, r1
1≤jr ≤pir ,1≤r≤t
t1
t pi
j1 ···jt t, r1
1≤jr ≤pir ,1≤r≤t
jr
t pi
t2k
k0
− −1
r
j1 ···jt t1k, r1
1≤jr ≤pir ,1≤r≤t
k0
t pi
−1t2k
r
jr
jr
−1
t1
t pi
j1 ···jt t, r1
1≤jr ≤pir ,1≤r≤t
r
jr
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pi1 ···pit −t
t pi
−1t1k
r
j1 ···jt tk, r1
1≤jr ≤pjr ,1≤r≤t
k0
jr
− −1t1
t pi
r
j1 ···jt t, r1
1≤jr ≤pir ,1≤r≤t
jr
−1t − −1t1 pi1 · · · pit .
3.15
Therefore, we have zi1 ,i2 ,...,it −1t pi1 · · · pit .
By Proposition 2.1 and Lemma 3.3, 3.11 becomes
RB4n 2RZ2n 2
1≤i1 <···<it ≤k,
1≤t≤k
2
−1
t1
1≤i1 <···<it ≤k,
1≤t≤k
−1t pi1 · · · pit R Z2n/pi1 ···pit
3.16
pi1 · · · pit R B4n/pi1 ···pit .
Let aβ1 ,...,βk : RB4n and let bβ1 ,...,βk : RZ2n . Then 3.16 becomes
aβ1 ,...,βk 2bβ1 ,...,βk 2
2
−1t pi1 · · · pit bπi1 ···it β1 ,...,βk 1≤i1 <···<it ≤k,
1≤t≤k
3.17
−1t1 pi1 · · · pit aπi1 ···it β1 ,...,βk .
1≤i1 <···<it ≤k,
1≤t≤k
Throughout the remaining part of the section, we solve the recurrence relation of
3.17 by using generating function technique. From now on, we allow each βi to be zero
for computational convenience.
Let
∞
∞
∞
βj
β βk−1
···
aβ1 ,...,βk xkk xk−1
· · · xj ,
ψβ1 ,...,βk xk , xk−1 , . . . , xj :
φβ1 ,...,βk xk , xk−1 , . . . , xj :
βj 0
βk−1 0 βk 0
∞
∞
∞
β βk−1
···
bβ1 ,...,βk xkk xk−1
βj 0
βk−1 0 βk 0
βj
· · · xj ,
3.18
where j k, k − 1, . . . , 1.
β β
β
For a fixed integer n p1 1 p2 2 · · · pkk such that p1 , . . . , pk are distinct prime numbers and
β1 , . . . , βk are non-negative integers, we define a function μ as follows.
μxk : 1 − 2pk xk ,
μ xk , . . . , xj : μ xk , . . . , xj1 − 1 μ xk , . . . , xj1 pj xj
for any j k − 1, k − 2, . . . , 1.
3.19
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Lemma 3.4. Let k be a positive integer. If k 1, then
μx1 ψβ1 x1 1 μx1 φβ1 x1 .
3.20
If k ≥ 2, then
μ xk , . . . , xj ψβ1 ,...,βk xk , . . . , xj
1 μ xk , . . . , xj
⎡
⎢
×⎢
⎣φβ1 ,...,βk xk , . . . , xj −1t pi1 · · · pit φπi1 ···it β1 ,...,βk xk , . . . , xj
1≤i1 <···<it ≤j−1,
1≤t≤j−1
3.21
⎤
⎥
−1t1 pi1 · · · pit ψπi1 ···it β1 ,...,βk xk , . . . , xj ⎥
⎦
1≤i1 <···<it ≤j−1,
1≤t≤j−1
for any j k, k − 1, . . . , 2.
Proof. Assume first that k 1. Then 3.17 with k 1 gives us that
aβ1 2bβ1 2p1 aβ1 −1 − 2p1 bβ1 −1 .
∞
Taking
β1 1
3.22
to both sides of 3.22, we have
1 − 2p1 x1 ψβ1 x1 2 − 2p1 x1 φβ1 x1 3.23
because a0 RB4p10 RZ22 22 and b0 RZ2p10 RZ2 2 by a direct computation.
From now on, we assume that k ≥ 2. We prove 3.21 by double induction on k and j.
Equation 3.17 with k 2 gives us that
aβ1 ,β2 2bβ1 ,β2 − 2p1 bβ1 −1,β2 − 2p2 bβ1 ,β2 −1 2p1 p2 bβ1 −1,β2 −1
2p1 aβ1 −1,β2 2p2 aβ1 ,β2 −1 − 2p1 p2 aβ1 −1,β2 −1 .
∞
Taking
β2 1
3.24
β
x2 2 of both sides of 3.24, we have
1 − 2p2 x2 ψβ1 ,β2 x2 2 − 2p2 x2 p1 ψβ1 −1,β2 x2 φβ1 ,β2 x2 − p1 φβ1 −1,β2 x2 3.25
because aβ1 ,0 aβ1 and bβ1 ,0 bβ1 by the definition, and
aβ1 ,0 − 2bβ1 ,0 − 2p1 aβ1 ,0 2p1 bβ1 −1,0 0
3.26
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Discrete Dynamics in Nature and Society
by 3.17 with k 1. That is,
μx2 ψβ1 ,β2 x2 1 μx2 φβ1 ,β2 x2 − p1 φβ1 −1,β2 x2 p1 ψβ1 −1,β2 x2 .
3.27
Thus 3.21 holds for k 2.
Assume now that 3.21 holds from 2 to k − 1 and consider the case for k. Note that the
last two terms of the right-hand side of 3.17 can be divided into three terms, respectively, as
follows:
2
−1t pi1 · · · pit bπi1 ···it β1 ,...,βk 1≤i1 <···<it ≤k,
1≤t≤k
−2pk bβ1 ,...,βk−1 ,βk −1 − 2pk
2
−1t pi1 · · · pit bπi1 ···it β1 ,...,βk−1 ,βk −1
1≤i1 <···<it ≤k−1,
1≤t≤k−1
−1t pi1 · · · pit bπi1 ···it β1 ,...,βk ,
1≤i1 <···<it ≤k−1,
1≤t≤k−1
2
3.28
−1t1 pi1 · · · pit aπi1 ···it β1 ,...,βk 1≤i1 <···<it ≤k,
1≤t≤k
2pk aβ1 ,...,βk−1 ,βk −1 − 2pk
2
−1t1 pi1 · · · pit aπi1 ···it β1 ,...,βk−1 ,βk −1
1≤i1 <···<it ≤k−1,
1≤t≤k−1
−1t1 pi1 · · · pit aπi1 ···it β1 ,...,βk .
1≤i1 <···<it ≤k−1,
1≤t≤k−1
Taking
∞
βk 1
β
xkk of both sides of 3.17 and using 3.28, one can see that
1 − 2pk xk ψβ1 ,...,βk xk 2 − 2pk xk
⎡
⎢
×⎢
⎣φβ1 ,...,βk xk −1t pi1 · · · pit φπi1 ···it β1 ,...,βk−1 ,βk xk 1≤i<···<it ≤k−1,
1≤t≤k−1
⎤
1≤i<···<it ≤k−1,
1≤t≤k−1
⎥
−1t1 pi1 · · · pit ψπi1 ···it β1 ,...,βk xk ⎥
⎦
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aβ1 ,...,βk−1 ,0 − 2bβ1 ,...,βk−1 ,0 − 2
−2
−1t pi1 · · · pit bπi1 ···it β1 ,...,βk−1 ,0
1≤i<···<it ≤k−1,
1≤t≤k−1
−1t1 pi1 · · · pit aπi1 ···it β1 ,...,βk−1 ,0 .
1≤i<···<it ≤k−1,
1≤t≤k−1
3.29
Further since
aβ1 ,...,βk−1 ,0 − 2bβ1 ,...,βk−1 ,0
−2
−1t pi1 · · · pit bπi1 ···it β1 ,...,βk−1 ,0
1≤i<···<it ≤k−1,
1≤t≤k−1
−2
3.30
−1t1 pi1 · · · pit aπi1 ···it β1 ,...,βk−1 ,0 0
1≤i<···<it ≤k−1,
1≤t≤k−1
by 3.17, we have
1 − 2pk xk ψβ1 ,...,βk xk 2 − 2pk xk
⎡
⎢
×⎢
⎣φβ1 ,...,βk xk 1≤i<···<it ≤k−1,
1≤t≤k−1
−1t pi1 · · · pit φπi1 ···it β1 ,...,βk−1 ,βk xk 3.31
⎤
⎥
−1t1 pi1 · · · pit ψπi1 ···it β1 ,...,βk xk ⎥
⎦.
1≤i<···<it ≤k−1,
1≤t≤k−1
Thus 3.21 holds for j k. Assume that 3.21 holds from k to j and consider the case for
j − 1. Note that the last two terms of the right-hand side of 3.21 can be divided into three
terms, respectively, as follows:
−1t pi1 · · · pit φπi1 ···it β1 ,...,βk xk , . . . , xj
1≤i1 <···<it ≤j−1,
1≤t≤j−1
−pj−1 φβ1 ,...,βj−2 ,βj−1 −1,βj ,...,βk xk , . . . , xj
− pj−1
−1t pi1 · · · pit φπi1 ···it β1 ,...,βj−2 ,βj−1 −1,βj ,...,βk xk , . . . , xj
1≤i1 <···<it ≤j−2,
1≤t≤j−2
−1t pi1 · · · pit φπi1 ···it β1 ,...,βk xk , . . . , xj ,
1≤i1 <···<it ≤j−2,
1≤t≤j−2
3.32
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−1t1 pi1 · · · pit ψπi1 ···it β1 ,...,βk xk , . . . , xj
1≤i1 <···<it ≤j−1,
1≤t≤j−1
pj−1 ψβ1 ,...,βj−2 ,βj−1 −1,βj ,...,βk xk , . . . , xj
− pj−1
−1t1 pi1 · · · pit ψπi1 ···it β1 ,...,βj−2 ,βj−1 −1,βj ,...,βk xk , . . . , xj
3.33
1≤i1 <···<it ≤j−2,
1≤t≤j−2
−1t1 pi1 · · · pit φπi1 ···it β1 ,...,βk xk , . . . , xj .
1≤i1 <···<it ≤j−2,
1≤t≤j−2
Taking
∞
βj−1 1
βj−1
xj−1 of both sides of 3.21, we have
μ xk , . . . , xj , xj−1 ψβ1 ,...,βk xj−1 , . . . , xk
1 μ xk , . . . , xj , xj−1
⎡
⎢
×⎢
⎣φβ1 ,...,βk xj−1 , . . . , xk −1t pi1 · · · pit φπi1 ···it β1 ,...,βk xk , . . . , xj , xj−1
1≤i1 <···<it ≤j−2,
1≤t≤j−2
⎤
⎥
−1t1 pi1 · · · pit ψπi1 ···it β1 ,...,βk xk , . . . , xj , xj−1 ⎥
⎦
1≤i1 <···<it ≤j−2,
1≤t≤j−2
μ xk , . . . , xj ψβ1 ,...,βj−2 ,0,βj ,...,βk xk , . . . , xj
− 1 μ xk , . . . , xj
−1t pi1 · · · pit φπi1 ···it β1 ,...,βj−2 ,0,βj ,...,βk xk , . . . , xj
1≤i1 <···<it ≤j−2,
1≤t≤j−2
− 1 μ xk , . . . , xj
−1t1 pi1 · · · pit ψπi1 ···it β1 ,...,βj−2 ,0,βj ,...,βk xk , . . . , xj .
1≤i1 <···<it ≤j−2,
1≤t≤j−2
3.34
Note that
μ xk , . . . , xj ψβ1 ,...,βj−2 ,0,βj ,...,βk xk , . . . , xj
− 1 μ xk , . . . , xj
−1t pi1 · · · pit φπi1 ···it β1 ,...,βj−2 ,0,βj ,...,βk xk , . . . , xj
1≤i1 <···<it ≤j−2,
1≤t≤j−2
− 1 μ xk , . . . , xj
1≤i1 <···<it ≤j−2,
1≤t≤j−2
−1t1 pi1 · · · pit ψπi1 ···it β1 ,...,βj−2 ,0,βj ,...,βk xk , . . . , xj 0
3.35
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by induction hypothesis. Thus
μ xk , . . . , xj , xj−1 ψβ1 ,...,βk xj−1 , . . . , xk
1 μ xk , . . . , xj , xj−1
⎡
⎢
×⎢
⎣φβ1 ,...,βk xj−1 , . . . , xk −1t pi1 · · · pit φπi1 ···it β1 ,...,βk xk , . . . , xj , xj−1
1≤i1 <···<it ≤j−2,
1≤t≤j−2
⎤
⎥
−1t1 pi1 · · · pit ψπi1 ···it β1 ,...,βk xk , . . . , xj , xj−1 ⎥
⎦.
1≤i1 <···<it ≤j−2,
1≤t≤j−2
3.36
Therefore, 3.21 holds for j − 1.
Equation 3.21 with j 2 gives us that
μxk , . . . , x2 ψβ1 ,...,βk xk , . . . , x2 1 μxk , . . . , x2 × φβ1 ,...,βk xk , . . . , x2 − p1 φβ1 −1,β2 ,...,βk xk , . . . , x2 p1 ψβ1 −1,β2 ,...,βk xk , . . . , x2 .
Taking
∞
β1 1
3.37
β
x1 1 of both sides of 3.37, we get that
μxk , . . . , x1 ψβ1 ,...,βk xk , . . . , x2 , x1 1 μxk , . . . , x1 φβ1 ,...,βk xk , . . . , x2 , x1 μxk , . . . , x2 ψ0,β2 ,...,βk xk , . . . , x2 − 1 μxk , . . . , x2 φ0,β2 ,...,βk xk , . . . , x2 .
3.38
Lemma 3.5. If k ≥ 2, then
μxk , . . . , x2 ψ0,β2 ,...,βk xk , . . . , x2 1 μxk , . . . , x2 φ0,β2 ,...,βk xk , . . . , x2 .
3.39
Proof. If k 2, then since ψ0,β2 x2 ψβ2 x2 and φ0,β2 x2 φβ2 x2 , the equation
μx2 ψ0,β2 x2 1 μx2 φ0,β2 x2 3.40
holds by 3.20. Assume now that 3.39 holds for k. Then by 3.38 we get that
μxk , . . . , x1 ψβ1 ,...,βk xk , . . . , x2 , x1 1 μxk , . . . , x1 φβ1 ,...,βk xk , . . . , x2 , x1 ,
3.41
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which implies that
μxk1 , . . . , x2 ψ0,β2 ,...,βk1 xk1 , . . . , x2 1 μxk1 , . . . , x2 φ0,β2 ,...,βk1 xk1 , . . . , x2 .
3.42
Thus 3.39 holds for k 1.
By Lemmas 3.4 and 3.5 and 3.38, we have
μxk , . . . , x1 ψβ1 ,...,βk xk , . . . , x1 1 μxk , . . . , x1 φβ1 ,...,βk xk , . . . , x1 .
3.43
We now need to find the function φβ1 ,...,βk xk , . . . , x1 explicitly.
Lemma 3.6. If p1 2, then
⎧ 2
⎪
,
if k 1,
⎪
⎪
⎨ λx1 φβ1 ,...,βk xk , . . . , x1 ⎪
⎪
1
1
⎪
⎩ 1
, if k ≥ 2.
λxk , . . . , x2 λxk , . . . , x1 3.44
If pi /
2 for i 1, 2, . . . , k, then
φβ1 ,...,βk xk , . . . , x1 1 1
1
.
λxk , . . . , x1 λxk , . . . , x1 3.45
Proof. We first assume that p1 2. Then by Proposition 2.2,
bβ1 ,β2 ,...,βk R Zpβ1 1 pβ2 pβ3 ···pβk
1
β 1 β
β
2
3
k
3.46
β
is the coefficient of x1 1 x2 2 x3 3 · · · xkk of
1
,
λxk , . . . , x1 β
β
3.47
β
β
which implies that bβ1 ,β2 ,...,βk is the coefficient of x1 1 x2 2 x3 3 · · · xkk of
1
2
λx1 if k 1,
1
1
λxk , . . . , x2 λxk , . . . , x1 3.48
if k ≥ 2,
Discrete Dynamics in Nature and Society
13
and hence by the definition of φ we get that
⎧ 2
⎪
,
⎪
⎪
⎨ λx1 φβ1 ,...,βk xk , . . . , x1 ⎪
⎪
1
1
⎪
⎩ 1
,
λxk , . . . , x2 λxk , . . . , x1 Assume now that pi /
2 for i 1, 2, . . . , k. Since bβ1 ,...,βk
Proposition 2.2 bβ1 ,...,βk is the coefficient of
β β
x11 x2 1 x3 2
βk
· · · xk1
if k 1,
3.49
if k ≥ 2.
RZ2pβ1 pβ2 ···pβk , by
1
2
k
of
1
.
λxk1 , . . . , x1 3.50
Since
1
1
λxk1 , . . . , x1 λxk1 , . . . , x2 − 1 λxk1 , . . . , x2 x1
3.51
1
1
λxk1 , . . . , x2 1 − 1 1/λxk1 , . . . , x2 x1
β
β
β
k
by the definition, bβ1 ,...,βk is the coefficient of x2 1 x3 2 · · · xk1
of
1
1
1
.
λxk1 , . . . , x2 λxk1 , . . . , x2 3.52
By changing the variables x2 , x3 , . . . , xk1 by x1 , x2 , . . . , xk , respectively, we get that bβ1 ,...,βk is
β
β
β
the coefficient of x1 1 x2 2 · · · xkk of
1
1
1
.
λxk , . . . , x1 λxk , . . . , x1 3.53
By the definition of φβ1 ,...,βk xk , . . . , x1 , we have
1
1
φβ1 ,...,βk xk , . . . , x1 1
.
λxk , . . . , x1 λxk , . . . , x1 3.54
By Proposition 2.1, 3.43, and Lemma 3.6, we have the following theorem.
Theorem 3.7. Let
β
β
β
n : p1 1 p2 2 · · · pkk ,
3.55
14
Discrete Dynamics in Nature and Society
be a positive integer such that p1 , . . . , pk are distinct prime numbers and β1 , . . . , βk are positive integers. Let
B4n : a, b | a2n e, b2 an , bab−1 a−1
3.56
be the dicyclic group of order 4n. Let RB4n be the number of rooted chains of subgroups in the lattice
of subgroups of B4n .
β
β
β
1 If p1 2, then RB4n is the coefficient of x1 1 x2 2 · · · xkk of
⎧
2
1
⎪
⎪
,
1
⎪
⎨
μx1 λx1 ψβ1 ,...,βk xk , . . . , x1 ⎪
1
1
1
⎪
⎪
⎩ 1
1
,
μxk , . . . , x1 λxk , . . . , x2 λxk , . . . , x1 β
β
if k 1,
if k ≥ 2.
3.57
β
2 If pi /
2 for i 1, 2, . . . , k, then RB4n is the coefficient of x1 1 x2 2 · · · xkk of
ψβ1 ,...,βk xk , . . . , x1 1 1
1
1
1
.
μxk , . . . , x1 λxk , . . . , x1 λxk , . . . , x1 3.58
Furthermore, the number CB4n of chains of subgroups in the lattice of subgroups of B4n is
β β
β
the coefficient of x1 1 x2 2 · · · xkk of
2ψβ1 ,...,βk xk , . . . , x1 −
β
Since
k
i1
1
.
1 − xi
β
3.59
β
We now want to find the coefficient of x1 1 x2 2 · · · xkk of ψβ1 ,...,βk xk , . . . , x1 explicitly.
1
1
1
,
μxk , . . . , x1 μxk , . . . , x2 1 − 1 1/μxk , . . . , x2 p1 x1
3.60
β
by the definition, the coefficient of x1 1 of 1/μxk , . . . , x1 is
β1
1
1
β
1
p1 1
μxk , . . . , x2 μxk , . . . , x2 i1 1
β1 1
β1 β1
p1
i1
μxk , . . . , x2 i1 0
i1 1
β1 1
β1 β1
p1
i1
μxk , . . . , x3 i1 0
3.61
1
1 − 1 1/μxk , . . . , x3 p2 x2
i1 1
.
Discrete Dynamics in Nature and Society
β
15
β
Thus the coefficient of x1 1 x2 2 of 1/μxk , . . . , x1 is
β
β
p1 1 p2 2
β1 β1
i1 β2
i1 0
β
i1
β2
1
μxk , . . . , x3 i1 1 1
β1 β2 β2
i1 β2
β2 β1
p1 1 p2
i1 0 i2 0
i1
i2
β2
1
μxk , . . . , x3 1
μxk , . . . , x3 β
β
β2
i1 i2 1
3.62
.
β
Continuing this process, one can see that the coefficient of x1 1 x2 2 · · · xkk of 1/μxk , . . . , x1 is
β β
2βk p1 1 p2 2
⎛
⎞
r
β
β1 β2
k−1 k−1 im ⎠
βr ⎝βr1 βk .
· · · pk
···
m1
ir
i1 0 i2 0
ik−1 0 r1
βr1
β
β
3.63
β
Similarly one can see that the coefficient of x1 1 x2 2 · · · xkk of 1/λxk , . . . , x1 is
2βk
β1 β2
i1 0 i2 0
β
β
···
⎛
⎞
r
im ⎠
βr ⎝βr1 ,
m1
ir
βr1
β
k−1 k−1 ik−1 0 r1
3.64
β
the coefficient of x1 1 x2 2 · · · xkk of 1 1/λxk , . . . , x2 1/λxk , . . . , x1 is
2βk
β
β2 β3
1 1 i1 0 i2 0 i3 0
β
β
⎛
⎞
r
k−1 im ⎠
β1 1
β2 i1 βr ⎝βr1 ···
m1
β2
ir
i1
r2
ik−1 0
βr1
β
k−1
3.65
β
and the coefficient of x1 1 x2 2 · · · xkk of 1/λxk , . . . , x1 2 is
⎞
⎛
r
β1 β
β2
k−1 k−1 i
1
β
βk m
βr ⎜ r1
⎟
···
β1 1 2
⎠.
⎝
m1
i
r
i1 0 i2 0
ik−1 0 r1
βr1
3.66
Therefore, one can have the following.
Corollary 3.8. Let n and B4n be the positive integer and the dicyclic group, respectively, defined in
Theorem 3.7. Let RB4n be the number of rooted chains of subgroups in the lattice of subgroups of
B4n .
16
Discrete Dynamics in Nature and Society
1 If p1 2, then
⎛
⎞
r
k−1 β
i
1
i
β
β
β
r1
m
1
2
1
r
⎝
⎠
RB4n 2βk
···
m1
β2
ir
i1
r2
i1 0 i2 0 i3 0
ik−1 0
βr1
⎛
⎞⎤
⎡⎡
r
β1 βk
jk−1 β2
j2
j1 k−1 j
i
j
j
j
j
r1
m
r ⎝
⎠⎦
⎣⎣p 1 p 2 · · · p k
2βk
···
···
m1
1 2
k
ir
j1 0 j 2 0
jk 0
i1 0 i2 0
ik−1 0 r1
jr1
⎡
β1
−j1 1 β
βk−1
−jk−1 2 −j2 β
3 −j3
β1 − j1 1
β2 − j2 i1
⎣
···
×
β2 − j2
i1
i 0 i 0 i 0
i 0
β
β2 β3
1 1 β
k−1
1
2
k−1
3
⎛
⎞⎤⎤
r
k−1 im ⎠
βr − jr ⎝βr1 − jr1 ⎦⎦,
×
m1
ir
r2
βr1 − jr1
3.67
where if k 1, then RB4·2β1 22β1 2 and if k 2, then
β
1 1
β1 1
β2 i1
β2
R B4·2β1 pβ2 2
β2
i1
2
i 0
1
β2
2
β1 β2
j1 0 j 2 0
j1 j1
j2 i1
j1 j2
2 p2
i1
j2
i1 0
⎡
β1
−j1 1
×⎣
i1 0
3.68
⎤⎤
β1 − j1 1
β2 − j2 i1 ⎦⎦
.
β2 − j2
i1
2 If pi / 2 for i 1, 2, . . . , k, then
RB4n 2βk
β1 β2
i1 0 i2 0
···
β
k−1 k−1 ik−1 0 r1
⎛
βr ⎝
ir
βr1 r
im
m1
⎞
⎠
βr1
⎛
⎞
r
β1 β
β2
k−1 k−1 βk im ⎠
βr ⎝βr1 1 β1 1 2
···
m1
ir
i1 0 i2 0
ik−1 0 r1
βr1
⎛
⎞⎤
⎡⎡
r
β1 βk
jk−1 β2
j2
j1 k−1 j
i
j
j
j
j
m⎠
r ⎝ r1
⎣⎣p 1 p 2 · · · p k
⎦
2βk
···
···
m1
1 2
k
i
r
j1 0 j2 0
jk 0
i1 0 i2 0
ik−1 0 r1
jr1
⎛
⎞⎤⎤
⎡
r
β
βk−1
−jk−1 1 −j1 β
2 −j2
k−1 β
−
j
i
βr − jr ⎝ r1 r1
m⎠
⎦⎦
×⎣
···
m1
i
r
i1 0 i2 0
ik−1 0 r1
βr1 − jr1
Discrete Dynamics in Nature and Society
β1 β2
βk
⎡⎡
⎣⎣pj1 pj2
2βk
···
1 2
j1 0 j 2 0
jk 0
17
⎛
⎞⎤
r
jk−1 j1 j2
k−1 im ⎠
jr ⎝jr1 jk ⎦
· · · pk
···
m1
ir
i1 0 i2 0
ik−1 0 r1
jr1
⎡
βk−1
−jk−1 1 −j1 β
2 −j2
k−1 β
βr − jr
⎣
× β1 − j1 1
···
ir
i 0 i 0
i 0 r1
1
k−1
2
⎛
⎞⎤⎤
r
βr1 − jr1 1 im ⎠
⎝
⎦⎦,
×
m1
βr1 − jr1
3.69
where if k 1, then
β1
β1
j
j R B4pβ1 2β1 β1 1 2β1 2β1 p11 2β1 p11 β1 − j1 1
1
j1 0
⎡
β 1
j1 0
β 2
p 1 − 1 p1 1
2β1 ⎣β1 2 1
p1 − 1
⎤
− β1 2 p1 β1 1
⎦.
2
p1 − 1
3.70
Acknowledgments
The first author was funded by the Korean Government KRF-2009-353-C00040. In the case
of the third author, this research was supported by Basic Science Research Program Through
the National Research Foundation of Korea NRF funded by the Ministry of Education,
Science and Technology 2011-0025252.
References
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2 V. Murali and B. B. Makamba, “Counting the number of fuzzy subgroups of an abelian group of order
pn qm ,” Fuzzy Sets and Systems, vol. 144, no. 3, pp. 459–470, 2004.
3 J.-M. Oh, “The number of chains of subgroups of a finite cycle group,” European Journal of Combinatorics,
vol. 33, no. 2, pp. 259–266, 2012.
4 M. Tărnăuceanu and L. Bentea, “On the number of fuzzy subgroups of finite abelian groups,” Fuzzy
Sets and Systems, vol. 159, no. 9, pp. 1084–1096, 2008.
5 J. M. Oh, “The number of chains of subgroups of the dihedral group,” Submitted.
6 J. S. Rose, A Course on Group Theory, Dover Publications, New York, NY, USA, 1994.
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