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I'm curious about the invariant factors of $(\mathbb{Z}/(1000))^\times$. I put down $$ (\mathbb{Z}/(1000))^\times\cong(\mathbb{Z}/(8))^\times\oplus(\mathbb{Z}/(125))^\times $$

It's easy to compute by hand that $(\mathbb{Z}/(8))^\times\cong\mathbb{Z}/(2)\oplus\mathbb{Z}/(2)$ and by a theorem of Gauss, $\mathbb{Z}/(125))^\times\cong\mathbb{Z}/(100)$ as it is cyclic.

Does this mean the list of invariant factors is just $(2,2,100)$?

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  • $\begingroup$ Sounds like an application of CRT. $\endgroup$
    – Debug
    Commented Sep 5 at 3:47

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Are you asking if the invariant factors of Z/2 x Z/2 x Z/100 are (2,2,100)? I think that is the definition of invariant factors, isn't it?

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