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

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Product of two cyclic groups is cyclic iff their orders are co-&#039;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

Jonas Dickerson · Accepted Answer

Step 1Note that&amp;nbsp;|G×H|=|G||H|=nm; so&amp;nbsp;G×H&amp;nbsp;is cyclic if and only if there is an element of order nm in&amp;nbsp;G×HIn any group A, if&amp;nbsp;a,&amp;nbsp;b∈A&amp;nbsp;commute with one another, a has order k, and b has order&amp;nbsp;ℓ&amp;nbsp;then the order of ab will divide&amp;nbsp;lcm(k,&amp;nbsp;ℓ)&amp;nbsp;(prove it).Now take an element of&amp;nbsp;G×H, written as&amp;nbsp;ga,&amp;nbsp;hb), where&amp;nbsp;G=⟨g⟩,&amp;nbsp;H=⟨h⟩,&amp;nbsp;0≤a&amp;lt;n,&amp;nbsp;0≤b&amp;lt;m. Then&amp;nbsp;(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 · Accepted Answer

Step 1BZm×Zn is noncyclic⇔ BZm×BZn⇔ {rmlcm}(m,n)&amp;lt;mn⇔ !gcd(m,n)&amp;gt;1

Product of two cyclic groups is cyclic iff their orders

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