Why is $Z_2 \times Z_3 \times Z_4$ not isomorphic to $Z_{24}$?

Question

Why is $Z_2 \times Z_3 \times Z_4$ not isomorphic to $Z_{24}$?

I had written this as a step in solving a problem on my math exam, and my teacher marked it as incorrect.

But I'm not sure as to why it's wrong, because $2$, $3$ and $4$ share no common factor?

Also, I'm sorry about the editing. I'm new to this site and not sure how to fix it.

Wow. I feel completely idiotic. I can't believe I made such a stupid mistake. Thank you so much for the clarification! — ConfusedSoul, Nov 04 '14 at 13:08

score 5 · Accepted Answer · answered Nov 04 '14 at 12:31

5

Certainly $2$ and $4$ share a common factor, namely $2$ itself.

Anyway, the groups are nonisomorphic, e.g., because $Z_{24}$ contains elements of order $24$ but the elements of $Z_2 \times Z_3 \times Z_4 \cong Z_2 \times Z_{12}$ all have order at most $12$.

answered Nov 04 '14 at 12:31

Travis Willse

99,363

Wow. I feel completely idiotic. I can't believe I made such a stupid mistake. Thank you so much for the clarification! – ConfusedSoul Nov 04 '14 at 13:07
1

It happens to the best of us. :) I'm glad you find my answer useful. – Travis Willse Nov 04 '14 at 13:13

score 0 · Answer 2 · answered Nov 04 '14 at 12:32

0

$\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_4$ is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_{12}$. But now $2, 12$ are not coprime, so you have that it is not isomorphic to $\mathbb{Z}_{24}$.

answered Nov 04 '14 at 12:32

Crostul

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Why is $Z_2 \times Z_3 \times Z_4$ not isomorphic to $Z_{24}$?

2 Answers2