MHB Understanding the Chinese Remainder Theorem for $\mathbb{Z}^{\times} _{20}$

[NoName3]

How do I show that $\mathbb{Z}^{\times} _{20} ≅ \mathbb{Z}_{2} \times \mathbb{Z}_{4}$?

I read that the chinese remainder theorem is the way to go but there are many versions and I can't find the right one. Most versions that I have found are statements between multiplicative groups, not from multiplicative group to additive product like we have here.

 

[I like Serena]

How do I show that $\mathbb{Z}^{\times} _{20} ≅ \mathbb{Z}_{2} \times \mathbb{Z}_{4}$?

I read that the chinese remainder theorem is the way to go but there are many versions and I can't find the right one. Most versions that I have found are statements between multiplicative groups, not from multiplicative group to additive product like we have here.

Hi NN!

Indeed, the Chinese Remainder Theorem says that:
$$\mathbb Z^\times_{20} \simeq \mathbb Z^\times_{2^2} \times \mathbb Z^\times_{5}$$
Is $\mathbb Z^\times_{2^2}$ isomorphic to $\mathbb{Z}_{2}$? (Wondering)

 

[NoName3]

Hi NN!

Indeed, the Chinese Remainder Theorem says that:
$$\mathbb Z^\times_{20} \simeq \mathbb Z^\times_{2^2} \times \mathbb Z^\times_{5}$$
Is $\mathbb Z^\times_{2^2}$ isomorphic to $\mathbb{Z}_{2}$? (Wondering)

Hi, I like Serena,

Thanks for the reply. Yes, I think they're isomorphic.

 

[I like Serena]

Hi, I like Serena,

Thanks for the reply. Yes, I think they're isomorphic.

So?

Oh, and why do you think they are isomorphic? (Wondering)

 

[NoName3]

So?

Oh, and why do you think they are isomorphic? (Wondering)

So $\mathbb Z^{\times}_{20} \simeq \mathbb Z_{2} \times \mathbb Z^\times_{5}$? As for why, two groups of the same order are isomorphic if $\gcd(n, \phi(n)) = 1$. This satisfies that.

 

[I like Serena]

So $\mathbb Z^{\times}_{20} \simeq \mathbb Z_{2} \times \mathbb Z^\times_{5}$?

Yep!

As for why, two groups of the same order are isomorphic if $\gcd(n, \phi(n)) = 1$. This satisfies that.

Huh?
I didn't know that yet, but it seems to be true.
Can you provide a reference?