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consider direct product of two finite groups, one is cyclic and the other one is not, is the direct product cyclic?

if both groups are not cyclic,what we can say about direct product of them?

I think if one of them is cyclic, the direct product is not cyclic,because for second group there is no generator and the generator of cyclic group doesn't effect on the second group,I think the same argument is right when both groups are non-cyclic,it will be great if you guide if I am wrong,thanks.

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  • $\begingroup$ Why don't you look at a direct product of, say, $\Bbb Z_2$ with a small symmetric group, or a small dihedral group? $\endgroup$ – pjs36 Apr 1 '15 at 15:48
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Any quotient group of a cyclic group is cyclic. So if you have a direct factor that is not cyclic, the group itself cannot be.

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For any finite group $G$, you can define the exponent $\exp G$ of $G$ to be the least natural number $N$ such that $g^N=e$ for all $g\in G$. It's clear that if $G$ is a cyclic group of order $n$, then $\exp G=n=\text{card}(G)$, in fact, the converse is also true: if $\exp G=\text{card}(G)$, then $G$ is a cyclic group (when $G$ is abelian, the proof is not so hard). Also it's easy to see that $\exp (G\times H)=\text{LCM}\ (\exp G,\exp H) $. From these properties, it can be proved that $G\times H$ is cyclic iff both $G$ and $H$ are cyclic and their orders are coprime.

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  • $\begingroup$ I wonder how this never got an upvote. +1 $\endgroup$ – Cauchy Aug 14 '17 at 7:38
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If the product of a cyclic group $C$ by another group $G$ is abelian, then $G$ is abelian, since it is a quotient of $C\times G$.

Similarly if the product of $C$ by $G$ is cyclic, $G$ is cyclic. This proves the product can be cyclic, nor abelian.

If both groups are not cyclic all that can be said is the product can't be either.

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