Discrete Dynamics in Nature and Society
Volume 2012, Issue 1 760246
Research Article
Open Access

The Number of Chains of Subgroups in the Lattice of Subgroups of the Dicyclic Group

Ju-Mok Oh, 

Ju-Mok Oh

Department of Mathematics, Kangnung-Wonju National University, Kangnung 210-702, Republic of Korea

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Yunjae Kim, 

Yunjae Kim

Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea donga.ac.kr

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Kyung-Won Hwang, 

Corresponding Author

Kyung-Won Hwang

Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea donga.ac.kr

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First published: 04 September 2012
https://doi.org/10.1155/2012/760246
Citations: 1
Academic Editor: Prasanta K. Panigrahi

Abstract

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 (L(G), ≤) where L(G) is the set of all subgroups of G and the partial order ≤ is the set inclusion. In this lattice (L(G), ≤), a chain of subgroups of G is a subset of L(G) 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.

2. Preliminaries

Given a group G, let 𝒞(G), 𝒰(G), and ℛ(G) 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 C(G): = |𝒞(G)|, U(G) : = |𝒰(G)|, and R(G): = |ℛ(G)|.

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 R(G) = U(G) + 1 and C(G) = R(G) + U(G) = 2R(G) − 1.

For a fixed positive integer k, we define a function λ as follows:
(2.1)
for any j = k − 1, k − 2, …, 1.

Proposition 2.2 (see [5].)Let   ℤn be the cyclic group of order

(2.2)
where p1, …, pk are distinct prime numbers and β1, …, βk are positive integers. Then the number R(ℤn) of rooted chains of subgroups in the lattice of subgroups of   ℤn is the coefficient of of
(2.3)

Let ℤ be the set of all integer numbers. Given distinct positive integers i1, …, it, we define a function
(2.4)
where
(2.5)

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].

3. The Number of Chains of Subgroups of the Dicyclic Group B4n

Throughout the section, we assume that
(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:
(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

(3.3)
which are isomorphic to the dicyclic group of order 4n/pi where i = 1,2, …, k.

Lemma 3.2. (1) For any i = 1,2, …, k,

(3.4)
where 0 ≤ r < s ≤ pi − 1.

(2) For any distinct prime factors of n,

(3.5)
where r1, …, rt are nonnegative integers.

Proof. (1) To the contrary suppose that

(3.6)
Then 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

(3.7)
Clearly, . Since , there exist integers u and v such that . Note that . On the other hand,
(3.8)
Considering the order of K, one can see that . Since
(3.9)
we have .

By Lemma 3.1, we have
(3.10)
Using the inclusion-exclusion principle and Lemma 3.2, one can see that the number U(B4n) has the following form:
(3.11)
for suitable integers and . In the following, we determine the numbers and explicitly.

Lemma 3.3. (1) .

(2) .

Proof. (1) Clearly 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 only appears in t-intersection of the subgroups

(3.12)
where and 1 ≤ r ≤ t. Since there are such choices, we have .

(2) By Lemma 3.2, one can see that among intersections of the subgroups of the right-hand side of (3.10), the group isomorphic to only appears one of the following two forms:

(3.13)
where 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 obtained from the first form, and let δ be the number of the groups isomorphic to obtained from the second form. Then clearly . Note that
(3.14)
On the other hand,
(3.15)
Therefore, we have .

By Proposition 2.1 and Lemma 3.3, (3.11) becomes
(3.16)
Let and let . Then (3.16) becomes
(3.17)

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
(3.18)
where j = k, k − 1, …, 1.
For a fixed integer such that p1, …, pk are distinct prime numbers and β1, …, βk are non-negative integers, we define a function μ as follows.
(3.19)
for any j = k − 1,   k − 2, …, 1.

Lemma 3.4. Let k be a positive integer. If k = 1, then

(3.20)
If k ≥ 2, then
(3.21)
for any j = k, k − 1, …, 2.

Proof. Assume first that k = 1. Then (3.17) with k = 1 gives us that

(3.22)
Taking to both sides of (3.22), we have
(3.23)
because and 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

(3.24)
Taking of both sides of (3.24), we have
(3.25)
because and by the definition, and
(3.26)
by (3.17) with k = 1. That is,
(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:

(3.28)
Taking of both sides of (3.17) and using (3.28), one can see that
(3.29)
Further since
(3.30)
by (3.17), we have
(3.31)
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:
(3.32)
(3.33)
Taking of both sides of (3.21), we have
(3.34)
Note that
(3.35)
by induction hypothesis. Thus
(3.36)
Therefore, (3.21) holds for j − 1.

Equation (3.21) with j = 2 gives us that
(3.37)
Taking of both sides of (3.37), we get that
(3.38)

Lemma 3.5. If k ≥ 2, then

(3.39)

Proof. If k = 2, then since and , the equation

(3.40)
holds by (3.20). Assume now that (3.39) holds for k. Then by (3.38) we get that
(3.41)
which implies that
(3.42)
Thus (3.39) holds for k + 1.

By Lemmas 3.4 and 3.5 and (3.38), we have
(3.43)
We now need to find the function explicitly.

Lemma 3.6. If p1 = 2, then

(3.44)
If pi ≠ 2 for i = 1,2, …, k, then
(3.45)

Proof. We first assume that p1 = 2. Then by Proposition 2.2,

(3.46)
is the coefficient of of
(3.47)
which implies that is the coefficient of of
(3.48)
and hence by the definition of ϕ we get that
(3.49)

Assume now that pi ≠ 2 for i = 1,2, …, k. Since , by Proposition 2.2 is the coefficient of of

(3.50)
Since
(3.51)
by the definition, is the coefficient of of
(3.52)
By changing the variables x2, x3, …, xk+1 by x1, x2, …, xk, respectively, we get that is the coefficient of of
(3.53)
By the definition of , we have
(3.54)

By Proposition 2.1, (3.43), and Lemma 3.6, we have the following theorem.

Theorem 3.7. Let

(3.55)
be a positive integer such that p1, …, pk are distinct prime numbers and β1, …, βk are positive integers. Let
(3.56)
be the dicyclic group of order 4n. Let R(B4n) be the number of rooted chains of subgroups in the lattice of subgroups of B4n.
  • (1)

    If p1 = 2, then R(B4n) is the coefficient of of

    (3.57)

  • (2)

    If pi ≠ 2 for i = 1,2, …, k, then R(B4n) is the coefficient of of

    (3.58)

Furthermore, the number C(B4n) of chains of subgroups in the lattice of subgroups of B4n is the coefficient of of

(3.59)

We now want to find the coefficient of of explicitly. Since
(3.60)
by the definition, the coefficient of of 1/μ(xk, …, x1) is
(3.61)
Thus the coefficient of of 1/μ(xk, …, x1) is
(3.62)
Continuing this process, one can see that the coefficient of of 1/μ(xk, …, x1) is
(3.63)
Similarly one can see that the coefficient of of 1/λ(xk, …, x1) is
(3.64)
the coefficient of of [1 + (1/λ(xk, …, x2))](1/λ(xk, …, x1)) is
(3.65)
and the coefficient of of 1/λ(xk, …, x1) 2 is
(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 R(B4n) be the number of rooted chains of subgroups in the lattice of subgroups of B4n.

  • (1)

    If p1 = 2, then

    (3.67)

where if k = 1, then and if k = 2, then
(3.68)
  • (2)

    If pi ≠ 2 for i = 1,2, …, k, then

    (3.69)

where if k = 1, then
(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).

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