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Discrete Dynamics in Nature and Society

VolumeΒ 2012Β (2012), Article IDΒ 760246, 17 pages

http://dx.doi.org/10.1155/2012/760246

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

^{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

Received 9 May 2012; Accepted 25 July 2012

Academic Editor: Prasanta K.Β Panigrahi

Copyright Β© 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 [1β4]). 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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