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American Journal of and Statistics 2018, 8(4): 96-98 DOI: 10.5923/j.ajms.20180804.03

On the Structure of Cyclic Groups of Order the Product of Distinct Primes

Rosemary Jasson Nzobo1,*, Benard Kivunge2, Waweru Kamaku3

1Pan African University Institute for Basic Sciences, Technology and Innovation, Nairobi, Kenya 2Department of Mathematics, Kenyatta University, Nairobi, Kenya 3Pure 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, at each level and number of ascending chains of 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 , Subgroups, Chains

it is isomorphic to for primes . In this paper, we present general formulas for finding the 1. Introduction 푛푚 number of levels, subgroupsℤ at each푛 ≠level푚 and number of A subgroup lattice is a diagram that includes all the ascending chains of cyclic groups of order … subgroups of the group and then connects a subgroup H at where are distinct primes. one level to a subgroup K at a higher level with a sequence 푝1푝2푝3 푝푚 of line segments if and only if H is a proper subgroup of K 푝푖 [1]. The study of subgroup lattice structures is traced back 2. Preliminaries from the first half of 20th century. For instance in 1953, Suzuki presented the extent to which a group is determined Definition 2.1 ([6]) Let L be a non empty set and < be a by its subgroup lattice in [2]. binary . In [2], Suzuki argued that isomorphic groups have the 1. A , poset ( , <) is called a lattice same lattice structure. Also, in [3], Birkhoff and Mac Lane if for every a,b in L, both sup { , } and { , } showed that up to isomorphism, there is only one cyclic belong to L. 퐿 group of order n. Hence for each n, there is exactly one 2. The lattice whose elements are the푎 푏 subgroups푖푛푓 of푎 the푏 subgroup lattice structure representing any of group G with the partial order relation being set order n. inclusion is called the subgroup lattice of the group G Jez in [1] deduced that the subgroup lattice structure of a and is denoted by ( ). cyclic group of prime power order (that is, when n = Definition 2.2 ([6]) Let G be a group. A sequence 퐿 퐺 where p is prime and k is a natural number) is a single chain.k of subgroups of G is called an In [1], it was also shown that if G is a and the푝 ascending chain. 1 2 3 subgroup lattice of G is a single chain, then G is cyclic. 퐻 Theorem⊆ 퐻 ⊆ 퐻 2.3⊆ ([3])⋯ 퐺 Up to Isomorphism, there is exactly That is, a finite group has a single chain subgroup lattice if one cyclic group of order n. and only if it is isomorphic to . Theorem 2.4 ([2]) Isomorphic groups have the same In [4], P'alfy showed that the subgroup lattice of a cyclic subgroup lattice diagram. 푛 group where n and m ℤare distinct primes has two Theorem 2.5 ([5]) If =< > is a cyclic group of ascending chains and three levels. Furthermore, P'alfy order n, then each subgroup of G has the form < > 푛푚 argued 퐶that any group whose subgroup lattice is formed by where d is a unique positive퐺 divisor푔 of n. 푑 two chains is isomorphic to . That is, a finite group has Theorem 2.6 ([5]) In a finite cyclic group, each subgroup푔 a subgroup lattice with two ascending chains if and only if has order dividing the order of the group. Conversely, given 푛푚 ℤ a positive of the order of the group, there is a * Corresponding author: subgroup of that order. [email protected] (Rosemary Jasson Nzobo) Theorem 2.7 ([5]) If =< > is a cyclic group of Published online at http://journal.sapub.org/ajms =< > =< > Copyright © 2018 The Author(s). Published by Scientific & Academic Publishing order n and , are subgroups of G This work is licensed under the Creative Commons Attribution International and are푑1 positive퐺 divisors푔 푑2 of n, then License (CC BY). http://creativecommons.org/licenses/by/4.0/ iff ( , 퐻) divides푔 퐻(′ , )푔. 푤ℎ푒푟푒 푑1 푑2 퐻 ⊆ 퐻′ 푔푐푑 푛 푑2 푔푐푑 푛 푑1 American Journal of Mathematics and Statistics 2018, 8(4): 96-98 97

3. Main Results For > 1, subgroups at the level are of the form . Since G is cyclic, given푡ℎ two subgroups of G, Theorem 3.1 If G is a cyclic group of order … 푟−푟1 and , 푟we have where are distinct primes and are positive푘1 푘2integers푘푚 푛⁄∏푖=1 푝푖 푚 푟−1 iff 푟= for = 1,2푟−, …1 . ( ) 푝1 푝2( ) 푝 푖=1 푖 푖=1 푖 푖=1 푖 and let be a subgroup lattice of G, then has Since푛⁄∏푟 푝 , and푟푛 there⁄∏ are푝 푟−11 distinct primes푛⁄∏ in푝 the⊆ 1 + 푝푖 푘푖 푖=1 푖 푖=1 푖 푗 푖=1 푖 . 푛product⁄∏ 푝푚 ,∏ we remain푝 푝 with∏ 푝 ( 푗1) = 푚+ 푚 퐿 퐺 퐿 퐺 푖=1 푖 Proof:푖=1 푖 Every subgroup of G is of the form 1 choices∏ of푟−푝 1 . Hence there are푟 − + 1 subgroups of G ∑ 푘 푖=1 푖 < … > where 0 . Subgroups at at ( + 1∏) level푝 such that 푚 − 푟 − 푚. − 푟 leve푥1l are푥2 such푥 푚that + + + = 1. The sum 푝푗 푚 − 푟 1 2 푚 푖 푖 Theorem푡ℎ 3.5: If G is a cyclic 푟group−1 of order푟 … of푡ℎ 푛these⁄푝 푝powers푝 at the first, second≤ 푥 ≤to 푘the last level are ⁄∏푖=1 푖 ⁄∏푖=1 푖 1 2 푚 where푟 are distinct primes 푛and ( 푝) ⊆is 푛a subgroup푝 lattice 푟0, 1, … respectively푥 푥 which⋯ 푥 form푟 − arithmetic 1 2 푚 of G, then the total number of chains ( ) from푝 푝 level푝 progression푚 with the first term = 0 , common 푖 푖=1 푖 to ( +푝1) level of ( ) is given퐿 by퐺 ( 푡ℎ+ 1). ∑ 푘 푟 difference = 1 and the last term = . The 푛 퐶 푟 1 푡ℎ 푚 ( + 1) number of levels,퐴푃 ( ) is the number of terms퐴 of푚 this . It Proof: The number of chains from 푟level−1 to 푛 푖=1 푖 푟 퐿 퐺 � � 푚 − 푟 follows that,푑 = 0 + ( ( ) 퐴1) which∑ 푘 gives level is the product of the number of subgroups푡ℎ at with푡ℎ 푟 푟 ( ) = 1 + 푛푚. 푙 퐴푃 the number of chains per subgroup from level푡ℎ to 푖=1 푖 푟 Remarks: Levels푚 ∑ are푘 counted from푛 푙 below,− that is from the ( + 1) level. From Theorem 3.3 and Lemma푡ℎ 3.4, the 푖=1 푖 푟 푛trivially푙 group∑ <푘0 > to the group G. number 푡ofℎ chains from level to ( + 1) level of ( ) 푟 ( ) 푡ℎ 푡ℎ Corollary 3.2 If G is a cyclic group of order … is then given by = ( + 1). 푟 푚 푟 퐿 퐺 where are distinct primes and ( ) is a subgroup lattice Theorem 3.6: If G푟 is a cyclic group of order … 1 2 푚 푛 퐶 �푟−1� 푚 − 푟 of G, then ( ) has + 1 levels. 푝 푝 푝 where are distinct primes and ( ) is a subgroup lattice 푖 1 2 푚 Proof:푝 Follows from Theorem 3.1퐿 퐺 where = 1 for all . of G, then the number of ascending chains 푝( 푝) from푝 푝푖 퐿 퐺 Theorem퐿 퐺3.3 If 푚G is a cyclic group of order = < 0 > to G is ! 푖 푛 퐶 … where are distinct primes 푘and ( ) is 푖a Proof: From Lemma 3.4, the number of chains per 푚 subgroup lattice of G, then the number ( ) of subgroups푛 subgroup from level to ( + 1) level is given by 1 2 푚 푖 푝a 푝 t푝 level of (푝) is given by ( ) = .퐿 퐺 + 1. To get푡ℎ the total number 푡ℎof ascending chains 푛 푆푟 from the trivial subgroup푟 to G is푟 simply taking products of Proof:푡ℎ For = 1 the result is trivially true푚 since the 푟 푟−1 푚these− 푟 numbers over all levels. It follows that, ( ) = trivial푟 group is the퐿 퐺only subgroup 푛at푆 the first� level� and we + 1 = ( 1)( 2) … 1 = ! have ( ) = 푟 = = 1. 푚Example 3.7: Let G be a cyclic group of order 푛=퐶5 × Subgroups at 푚the second푚 level ( = 2) are of the form ∏푟=1 푚 − 푟 푚 푚 − 푚 − 푚 푛 푆1 �1−1� � 0 � 7 × 13 × 19, here = 4. The lattice of G is as shown < > for = 1,2, … . Since for each prime there below; 푛 is a unique subgroup < > of 푟G, the total number of 푖 푖 푚 subgroups푛⁄푝 of this푖 form is푚 the number of ways of choosing푝 푖 (without repetition) one 푛prime⁄푝 from m primes. Hence there are = subgroups at the second level. 푖 푚 푚 푝 ( = 3) Again, subgroups1 2−1 at the third level are of the form < � � �> for� , = 1,2, … . Since for each <푟 > product 푖there푗 is a unique subgroup of G, the total number푛⁄푝 푝 of subgroups푖 푗 of this 푚form is the number of 푖 푗 푖 푗 ways of choosing푝 푝 (without repetition) two primes푛⁄푝 푝 from m primes. Hence there are = subgroups. 푝푖푝푗 In a similar way, subgroups푚 at the 푚 level are generated � 2 � �3−1� by of the form . Since푡ℎ there is a unique subgroup for each divisor of 푟this−1 form,푟 we choose (without 푖=1 푖 repetition) 1 primes푛 ⁄∏ from푝 primes hence there are subgroups at the . 푖 푚 푟 − 푡푝ℎ 푚 = … 푟−Lemma1 3.4 If G is a cyclic group of order �where� are distinct primes푟 and ( ) is a subgroup lattice 1 2 푚 Figure 1. The lattice of a cyclic group G of G, then for every subgroup H of G at 푛level,푝 there푝 푝are 푖 +푝 1 chains towards ( + 1퐿) 퐺 level.푡ℎ From Figure 1, we see that the lattice of G has five levels Proof: At the first level ( = 1푡ℎ), there푟 is only trivial which is equivalent to + 1 = 4 + 1 . This justifies 푚group− 푟 which is the subgroup of푟 all m subgroups of the second Corollary 3.2. level. But we also have 1푟+ 1 = hence the result is To justify Theorem 3.3 푚for the number of subgroups at true for =l. each level, we use the following table. 푚 − 푚 푟

98 Rosemary Jasson Nzobo et al.: On the Lattice Structure of Cyclic Groups of Order the Product of Distinct Primes

Table 1. Number of Subgroups at Each Level 4. Conclusions Level ( ) Subgroups at ( ¦ 1) General formulas for number of levels, the number of 1 푟 1 퐺 푚(4¦푟0−)=1 subgroups at each level and the number of ascending chains 2 4 (4¦1)=4 are presented. It was proved that the number of subgroups at the level of a lattice structure ( ) of a cyclic group of 3 6 (4¦2)=6 order 푡ℎthe product of m distinct primes is given by m 4 4 (4¦3)=4 combination푟 1. Furthermore,퐿 it퐺 was proved that the 5 1 (4¦4)=1 number of ascending chains of ( ) is given by ! In future work,푟 − we will study the lattice structure of cyclic We can also see that there are 24 ascending chains which groups of order the product of퐿 m퐺 prime powers.푚 We will is equivalent to ! = 4! as stated in Theorem 3.6. These deduce general formulas for finding the number of chains are: subgroups at each level and number of ascending chain. < 0 > <푚5.7.13 > < 5.7 > < 5 > 1. ------2. < 0 >--< 5.7.13 >--< 5.7 >--< 7 >-- 3. < 0 >--< 5.7.13 >--< 5.13 >--< 5 >-퐺 4. < 0 >--< 5.7.13 >--< 5.13 >-< 13 >퐺- REFERENCES 5. < 0 >--< 5.7.13 >--< 7.13 >--< 7 >--퐺 퐺 [1] A. Jez, Subgroup Lattices That Are Chains, Rose-Hulman 6. < 0 >--< 5.7.13 >--< 7.13 >--< 13 >-- Undergraduate Mathematics Journal, vol. 70.2 p. 4, 2006. 7. < 0 >--< 5.7.19 >--< 5.7 >--< 5 >-- 퐺 [2] M. Suzuki, On the lattice of subgroups of finite groups, 8. < 0 >--< 5.7.19 >--< 5.7 >--< 7 >-- 퐺 퐺 Transactions of the American Mathematical Society, vol. 9. < 0 >--< 5.7.19 >--< 5.19 >--< 5 >- 70.2, pp 345 – 371, 1951. 10. < 0 >--< 5.7.19 >--< 5.19 >-< 19 >퐺- 퐺 [3] G. Birkhoff and S.M. Lane, A survey of modern algebra, 11. < 0 >--< 5.7.19 >--< 7.19 >--< 7 >-- Universities Press, 1958. 12. < 0 >--< 5.7.19 >--< 7.19 >--< 19 >퐺-- 퐺 [4] P.P. P'alfy, Groups and lattices, Groups St Andrews 2001 in 13. < 0 >--< 5.13.19 >--< 5.13 >--< 5 >-- Oxford, vol. 305 pp. 429 – 454. 14. < 0 >--< 5.13.19 >--< 5.13 >--< 13 >--퐺 [5] K. Conrad, Subgroups of cyclic groups. www.math.uconn.ed 15. < 0 >--< 5.13.19 >--< 5.19 >--< 5 >--퐺 퐺 u/~kconrad/blurbs/grouptheory/cyclicgp.pdf. 16. < 0 >--< 5.13.19 >--< 5.19 >-< 19 >-- 17. < 0 >--< 5.13.19 >--< 13.19 >--< 13 >퐺-- [6] R. Singh, Cyclic groups, Researchgate, 2016. 퐺 18. < 0 >--< 5.13.19 >--< 13.19 >--< 19 >-- 퐺 19. < 0 >--< 7.13.19 >--< 7.13 >--< 7 >-- 퐺 20. < 0 >--< 7.13.19 >--< 7.13 >--< 13 >-- 21. < 0 >--< 7.13.19 >--< 7.19 >--< 7 >--퐺 22. < 0 >--< 7.13.19 >--< 7.19 >-< 19 >-- 퐺 23. < 0 >--< 7.13.19 >--< 13.19 >--< 13 >퐺-- 24. < 0 >--< 7.13.19 >--< 13.19 >--< 19 >퐺-- 퐺 퐺