Product of two cyclic groups is cyclic iff their orders

coraletsmmh

coraletsmmh

Answered question

2022-04-23

Product of two cyclic groups is cyclic iff their orders are co-'
Say you have two groups G=g with order n and H=h with order m. Then the product G×H is a cyclic group if and only if gcd(n,m)=1

Answer & Explanation

Jonas Dickerson

Jonas Dickerson

Beginner2022-04-24Added 22 answers

Step 1
Note that |G×H|=|G||H|=nm; so G×H is cyclic if and only if there is an element of order nm in G×H
In any group A, if a, bA commute with one another, a has order k, and b has order  then the order of ab will divide lcm(k, ) (prove it).
Now take an element of G×H, written as ga, hb), where G=g, H=h, 0a<n, 0b<m. Then (ga,hb)=(ga,1)(1,hb)
In this case, what is the order? Under what conditions can you get an element of order exactly nm, which is what you need

drenkttj9

drenkttj9

Beginner2022-04-25Added 20 answers

Step 1
BZm×Zn is noncyclic
 BZm×BZn
 {rmlcm}(m,n)<mn
 !gcd(m,n)>1

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