American Journal of Mathematics and Statistics 2018, 8(4): 96-98
DOI: 10.5923/j.ajms.20180804.03
On the Lattice Structure of Cyclic Groups of Order
the Product of Distinct Primes
Rosemary Jasson Nzobo
1,*
, Benard Kivunge
2
, Waweru Kamaku
3
1
Pan African University Institute for Basic Sciences, Technology and Innovation, Nairobi, Kenya
2
Department of Mathematics, Kenyatta University, Nairobi, Kenya
3
Pure and Applied Mathematics Department, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Abstract In this paper, we give general formulas for counting the number of levels, subgroups at each level and number
of ascending chains of subgroup lattice of cyclic groups of order the product of distinct primes. We also give an example to
illustrate the concepts introduced in this work.
Keywords Lattice, Cyclic Group, Subgroups, Chains
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 n=
k
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 ℤ
.
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 ℤ
. That is, a finite group has
a subgroup lattice with two ascending chains if and only if
* Corresponding author:
nzobor@gmail.com (Rosemary Jasson Nzobo)
Published online at http://journal.sapub.org/ajms
Copyright © 2018 The Author(s). Published by Scientific & Academic Publishing
This work is licensed under the Creative Commons Attribution International
License (CC BY). http://creativecommons.org/licenses/by/4.0/
it is isomorphic to ℤ
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
1
2
3
…
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 sup {, } 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
1
⊆
2
⊆
3
⊆⋯ 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 =<
1
>, ′ =<
2
>are subgroups of G
ℎ
1
and
2
are positive divisors of n, then ⊆ ′
iff (,
2
) divides (,
1
).