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An abelian group is expressed into the canonical form:

$$\bigoplus\mathbb{Z}_{k_i}$$

where $k_i$ are powers of primes. I want to know if the group is cyclic or not.

It seems from what I've read that such a group is cyclic if and only if $\forall i\neq j: \gcd(k_i,k_j)=1$.

I even managed to find an answer that explicitly states this. However the questions and answers that I could find are mainly concerned with the "if" part. I understand that one. But is the "only if" part true? How to prove that the group is not cyclic if some pair of the $k_i$'s are powers of the same prime?

Džuris
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1 Answers1

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If $\gcd(a,b)=1$ then $\Bbb Z_a\oplus \Bbb Z_b\cong \Bbb Z_{ab}$. So we can bunch together coprime cyclic factors, and will succeed in getting a cyclic group unless we get a pair of factors $\Bbb Z_a\oplus \Bbb Z_b$ with $g=\gcd(a,b)>1$. But in that case $\Bbb Z_a\oplus \Bbb Z_b$ has $g^2$ solutions to $gx=0$, so it isn't cyclic, and neither can any group containing it be cyclic.

Angina Seng
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  • Or $\Bbb Z_a\oplus \Bbb Z_b$ has exponent $ab/g < ab$ and so is not cyclic. – lhf Apr 24 '17 at 18:07
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    Sorry, I must be missing something very basic. I managed to follow the argument all up to "so it isn't cyclic". Yes I see that there are $g^2$ solutions for $gx=0$. How does not being cyclic follow from that? – Džuris Apr 24 '17 at 18:32