1. ## Torsion

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If

2. ## Re: Torsion

Originally Posted by jcir2826
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If
I don't know what i) says, presumably that $\displaystyle \displaystyle \text{Tor}(G)= \bigoplus_{p}\mathbb{Z}_p$--this is easy. To prove ii) prove that for every $\displaystyle \displaystyle (x_p)\in G$ and $\displaystyle n=p_1^{\alpha_1}\cdots p_n^{\alpha_n}$ you can find $\displaystyle \displaystyle (y_p)$ such that $\displaystyle (x_p)-n(y_p)\in\text{Tor}(G)$. To do this factorize $\displaystyle n=p_1^{\alpha_1}\cdots p_m^{\alpha_m}$. Then, note that for $\displaystyle p\ne p_1,\cdpts,p_m^{\alpha}_m$ you have that $\displaystyle n\in\mathbb{Z}_p^\times$. Define then $\displaystyle (y_p)$ to be such that $\displaystyle y_p=0$ if $\displaystyle p\in\{p_1,\cdots,p_m\}$ and $\displaystyle [n]_p^{-1}x_p$ otherwise. To prove three note that if $\displaystyle G/\text{Tor}(G)$ were a direct summand of $\displaystyle G$ then there would be an exact sequence $\displaystyle 0\to G/\text{Tor}(G)\to G$, and what's the problem with that (hint: $\displaystyle G$ is HIGHLY non-divisible).