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Prove that two given groups are nonisomorphic

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.13

Prove that the following pairs of groups are not isomorphic:

(1) $\mathbb{Z} \times Z_2$ and $\mathbb{Z}$,

(2) $\mathbb{Q} \times Z_2$ and $\mathbb{Q}$.


(1) We saw in Exercise 2.3.12 that $\mathbb{Z} \times Z_2$ is not cyclic, but $\mathbb{Z}$ is cyclic. So these cannot be isomorphic.

(2) Note that no nonidentity element of $\mathbb{Q}$ has finite order. However, $(0,x)$ has order 2 in $\mathbb{Q} \times Z_2$; these groups cannot be isomorphic.


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