Showing the injectivity of the following homomorphism (direct product of normal Sylow subgroups)

Question

Consider normal Sylow subgroups $G_1, \ldots, G_r$ of a finite abelian group $G$. Let's set up the homomorphism $\phi: G_1 \times \ldots \times G_r \to G$ s.t. $(g_1, \ldots, g_r) \mapsto g_1\ldots g_r$ as in this question. How to show that the homomorphism is injective, i.e., the kernel is trivial?

I understand that $g_1, \ldots, g_n$ raised to their respective orders is certainly in the kernel but I'm not sure how to show that no other $r$-tuple is in the kernel too. I think the fact that "in abelian groups, the order product of commuting elements with relatively prime orders is equal to the product of their orders" needs to be used somehow but I'm not sure how.

Answer 1

Let $|G_{i}|=p_{i}^{n_{i}}$ for all $i$. Let $(g_{1},\ldots,g_{r})\in\ker\phi$. Then $g_{1}\cdots g_{r}=1$. If $r=1$, then we are done. If $r\geq 2$, for any $i$, we have $g_{i}=(g_{1}\cdots g_{i-1}g_{i+1}\cdots g_{r})^{-1}$. Since $g_{i}^{p_{i}^{n_{i}}}=1$, $|g_{i}|\mid p_{i}^{n_{i}}$. Since $$ g_{i}^{p_{1}^{n_{1}}\cdots p_{i-1}^{n_{i-1}}p_{i+1}^{n_{i+1}}\cdots p_{r}^{n_{r}}}=(g_{1}\cdots g_{i-1}g_{i+1}\cdots g_{r})^{-p_{1}^{n_{1}}\cdots p_{i-1}^{n_{i-1}}p_{i+1}^{n_{i+1}}\cdots p_{r}^{n_{r}}}=1, $$ $|g_{i}|\mid p_{1}^{n_{1}}\cdots p_{i-1}^{n_{i-1}}p_{i+1}^{n_{i+1}}\cdots p_{r}^{n_{r}}$. So $|g_{i}|\mid \gcd(p_{i}^{n_{i}},p_{1}^{n_{1}}\cdots p_{i-1}^{n_{i-1}}p_{i+1}^{n_{i+1}}\cdots p_{r}^{n_{r}})=1$. Then $g_{i}=1$. So $\phi$ is injective.