Find a product of cyclic groups isomorphic to Z/20

Question

I have to decompose the multiplicative group Z/20 into a product of cyclic groups. Now the order of Z/20 is $\phi(20) = 8$ right? So by the fundamental theorem of abelian groups I factorize $8$ into $2^3$ and hence Z/20 is isomorphic to either Z/8, Z/4 x Z/2 or Z/2 x Z/2 x Z/2. Is that correct or have I made a mistake somewhere? Any tips are welcome

Answer 1

You seem to have confused $\mathbb Z_{20}$ and $\mathbb Z_{20}^\times$. The former has order $20$, the latter $\varphi(20)=8$.

Let's say you are talking about the former. Then you can use CRT to write it as a product of cyclic groups. Then $\mathbb Z_{20}\cong\mathbb Z_4\times\mathbb Z_5$.

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On the other hand, for the slightly more difficult problem, $\mathbb Z_{20}^×\cong (\mathbb Z_4×\mathbb Z_5)^×\cong\mathbb Z_4^××\mathbb Z_5^×\cong\mathbb Z_2×\mathbb Z_4$, because the group of units functor respects products.