Discrete Dynamics in Nature and Society

Volume 2012 (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

¹Department of Mathematics, Kangnung-Wonju National University, Kangnung 210-702, Republic of Korea

²Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Received 9 May 2012; Accepted 25 July 2012

Academic Editor: Prasanta K. Panigrahi

Copyright © 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.

Abstract.

We give an explicit formula for the number of chains of subgroups in the lattice of subgroups of the dicyclic group of order 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 is the lattice where is the set of all subgroups of and the partial order is the set inclusion. In this lattice , a chain of subgroups of is a subset of linearly ordered by set inclusion. A chain of subgroups of is called -rooted (or rooted) if it contains . Otherwise, it is called unrooted.

The problem of counting chains of subgroups of a given group has received attention by researchers with related to classifying fuzzy subgroups of 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 of order where 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 of order by finding its generating function of multivariables where is an arbitrary integer.

2. Preliminaries

Given a group , let , , and be the collection of chains of subgroups of , of unrooted chains of subgroups of , and of -rooted chains of subgroups of , respectively. Let , , and .

The following simple observation is useful for enumerating chains of subgroups of a given group.

Proposition 2.1. Let be a finite group. Then and .

For a fixed positive integer , we define a function as follows:

for any .

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

where are distinct prime numbers and are positive integers. Then the number of rooted chains of subgroups in the lattice of subgroups of is the coefficient of of

Let be the set of all integer numbers. Given distinct positive integers , we define a function

where

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

Throughout the section, we assume that

is a positive integer, where are distinct prime numbers and are nonnegative integers and the dicyclic group of order is defined by the following presentation:

where is the identity element.

By the elementary group theory, the following is wellknown.

Lemma 3.1. The dicyclic group has an index subgroup , which is isomorphic to , and has index subgroups

which are isomorphic to the dicyclic group of order where .

Lemma 3.2. (1) For any ,

where .

(2) For any distinct prime factors of ,

where are nonnegative integers.

Proof. (1) To the contrary suppose that

Then for some integers and . This implies . Since , we have , a contradiction.

(2) We only give its proof when . The general case can be proved by the inductive process. Let

Clearly, . Since , there exist integers and such that . Note that . On the other hand,

Considering the order of , one can see that . Since

we have .

By Lemma 3.1, we have

Using the inclusion-exclusion principle and Lemma 3.2, one can see that the number has the following form:

for suitable integers and . In the following, we determine the numbers and explicitly.

Lemma 3.3. (1) .

(2) .

Proof. (1) Clearly for any . For any integer , 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 -intersection of the subgroups

where and . 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:

where and , 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 be the number of the groups isomorphic to obtained from the first form, and let be the number of the groups isomorphic to obtained from the second form. Then clearly . Note that

On the other hand,

Therefore, we have .

By Proposition 2.1 and Lemma 3.3, (3.11) becomes

Let and let . Then (3.16) becomes

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 to be zero for computational convenience.

Let

where .

For a fixed integer such that are distinct prime numbers and are non-negative integers, we define a function as follows.

for any .

Lemma 3.4. Let be a positive integer. If , then

If , then

for any .

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

Taking to both sides of (3.22), we have

because and by a direct computation.

From now on, we assume that . We prove (3.21) by double induction on and . Equation (3.17) with gives us that

Taking of both sides of (3.24), we have

because and by the definition, and

by (3.17) with . That is,

Thus (3.21) holds for .

Assume now that (3.21) holds from to and consider the case for . Note that the last two terms of the right-hand side of (3.17) can be divided into three terms, respectively, as follows:

Taking of both sides of (3.17) and using (3.28), one can see that

Further since

by (3.17), we have

Thus (3.21) holds for . Assume that (3.21) holds from to and consider the case for . Note that the last two terms of the right-hand side of (3.21) can be divided into three terms, respectively, as follows:

Taking of both sides of (3.21), we have

Note that

by induction hypothesis. Thus

Therefore, (3.21) holds for .

Equation (3.21) with gives us that

Taking of both sides of (3.37), we get that

Lemma 3.5. If , then

Proof. If , then since and , the equation

holds by (3.20). Assume now that (3.39) holds for . Then by (3.38) we get that

which implies that

Thus (3.39) holds for .

By Lemmas 3.4 and 3.5 and (3.38), we have

We now need to find the function explicitly.

Lemma 3.6. If , then

If for , then

Proof. We first assume that . Then by Proposition 2.2,

is the coefficient of of

which implies that is the coefficient of of

and hence by the definition of we get that

Assume now that for . Since , by Proposition 2.2 is the coefficient of of

Since

by the definition, is the coefficient of of

By changing the variables by , respectively, we get that is the coefficient of of

By the definition of , we have

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

Theorem 3.7. Let

be a positive integer such that are distinct prime numbers and are positive integers. Let

be the dicyclic group of order . Let be the number of rooted chains of subgroups in the lattice of subgroups of .

(1)If , then is the coefficient of of

(2)If for , then is the coefficient of of

Furthermore, the number of chains of subgroups in the lattice of subgroups of is the coefficient of of

We now want to find the coefficient of of explicitly. Since

by the definition, the coefficient of of is

Thus the coefficient of of is

Continuing this process, one can see that the coefficient of of is

Similarly one can see that the coefficient of of is

the coefficient of of is

and the coefficient of of is

Therefore, one can have the following.

Corollary 3.8. Let and be the positive integer and the dicyclic group, respectively, defined in Theorem 3.7. Let be the number of rooted chains of subgroups in the lattice of subgroups of .

(1)If , then

where if , then and if , then

(2)If for , then

where if , then

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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