1. Introduction

A subgroup lattice is a diagram that includes all the subgroups of the group and then connects a subgroup H at one level to a subgroup K at a higher level with a sequence of line segments if and only if H is a proper subgroup of K [1]. The study of subgroup lattice structures is traced back from the first half of 20^th century. For instance in 1953, Suzuki presented the extent to which a group is determined by its subgroup lattice in [2].

In [2], Suzuki argued that isomorphic groups have the same lattice structure. Also, in [3], Birkhoff and Mac Lane showed that up to isomorphism, there is only one cyclic group of order n. Hence for each n, there is exactly one subgroup lattice structure representing any cyclic group of order n.

Jez in [1] deduced that the subgroup lattice structure of a cyclic group of prime power order (that is, when where p is prime and k is a natural number) is a single chain. In [1], it was also shown that if G is a finite group and the subgroup lattice of G is a single chain, then G is cyclic. That is, a finite group has a single chain subgroup lattice if and only if it is isomorphic to

In [4], P'alfy showed that the subgroup lattice of a cyclic group where n and m are distinct primes has two ascending chains and three levels. Furthermore, P'alfy argued that any group whose subgroup lattice is formed by two chains is isomorphic to That is, a finite group has a subgroup lattice with two ascending chains if and only if it is isomorphic to for primes

In this paper, we present general formulas for finding the number of levels, subgroups at each level and number of ascending chains of cyclic groups of order where are distinct primes.

2. Preliminaries

Definition 2.1 ([6]) Let L be a non empty set and < be a binary relation.

1. A partially ordered set, poset is called a lattice if for every a,b in L, both and belong to L.

2. The lattice whose elements are the subgroups of the group G with the partial order relation being set inclusion is called the subgroup lattice of the group G and is denoted by .

Definition 2.2 ([6]) Let G be a group. A sequence of subgroups of G is called an ascending chain.

Theorem 2.3 ([3]) Up to Isomorphism, there is exactly one cyclic group of order n.

Theorem 2.4 ([2]) Isomorphic groups have the same subgroup lattice diagram.

Theorem 2.5 ([5]) If is a cyclic group of order n, then each subgroup of G has the form where d is a unique positive divisor of n.

Theorem 2.6 ([5]) In a finite cyclic group, each subgroup has order dividing the order of the group. Conversely, given a positive divisor of the order of the group, there is a subgroup of that order.

Theorem 2.7 ([5]) If is a cyclic group of order n and are subgroups of G where and are positive divisors of n, then iff divides .

3. Main Results

Theorem 3.1 If G is a cyclic group of order where are distinct primes and are positive integers and let be a subgroup lattice of G, then has .

Proof: Every subgroup of G is of the form where Subgroups at level are such that The sum of these powers at the first, second to the last level are respectively which form arithmetic progression with the first term , common difference and the last term The number of levels, is the number of terms of this It follows that, which gives

Remarks: Levels are counted from below, that is from the trivially group to the group G.

Corollary 3.2 If G is a cyclic group of order where are distinct primes and is a subgroup lattice of G, then has levels.

Proof: Follows from Theorem 3.1 where for all

Theorem 3.3 If G is a cyclic group of order where are distinct primes and is a subgroup lattice of G, then the number of subgroups a level of is given by

Proof: For the result is trivially true since the trivial group is the only subgroup at the first level and we have

Subgroups at the second level are of the form for Since for each prime there is a unique subgroup of G, the total number of subgroups of this form is the number of ways of choosing (without repetition) one prime from m primes. Hence there are subgroups at the second level.

Again, subgroups at the third level are of the form for Since for each product there is a unique subgroup of G, the total number of subgroups of this form is the number of ways of choosing (without repetition) two primes from m primes. Hence there are subgroups.

In a similar way, subgroups at the level are generated by divisors of the form Since there is a unique subgroup for each divisor of this form, we choose (without repetition) primes from primes hence there are subgroups at the .

Lemma 3.4 If G is a cyclic group of order where are distinct primes and is a subgroup lattice of G, then for every subgroup H of G at level, there are chains towards level.

Proof: At the first level there is only trivial group which is the subgroup of all m subgroups of the second level. But we also have hence the result is true for

For subgroups at the level are of the form Since G is cyclic, given two subgroups of G, and we have iff for Since and there are distinct primes in the product we remain with choices of Hence there are subgroups of G at level such that

Theorem 3.5: If G is a cyclic group of order where are distinct primes and is a subgroup lattice of G, then the total number of chains from level to level of is given by

Proof: The number of chains from level to level is the product of the number of subgroups at with the number of chains per subgroup from level to level. From Theorem 3.3 and Lemma 3.4, the number of chains from level to level of is then given by

Theorem 3.6: If G is a cyclic group of order where are distinct primes and is a subgroup lattice of G, then the number of ascending chains from to G is

Proof: From Lemma 3.4, the number of chains per subgroup from level to level is given by To get the total number of ascending chains from the trivial subgroup to G is simply taking products of these numbers over all levels. It follows that,

Example 3.7: Let G be a cyclic group of order here The lattice of G is as shown below;

Figure 1. The lattice of a cyclic group G

From Figure 1, we see that the lattice of G has five levels which is equivalent to This justifies Corollary 3.2.

To justify Theorem 3.3 for the number of subgroups at each level, we use the following table.

Table 1. Number of Subgroups at Each Level

We can also see that there are 24 ascending chains which is equivalent to as stated in Theorem 3.6. These chains are:

4. Conclusions

General formulas for number of levels, the number of subgroups at each level and the number of ascending chains are presented. It was proved that the number of subgroups at the level of a lattice structure of a cyclic group of order the product of m distinct primes is given by m combination Furthermore, it was proved that the number of ascending chains of is given by

In future work, we will study the lattice structure of cyclic groups of order the product of m prime powers. We will deduce general formulas for finding the number of subgroups at each level and number of ascending chain.