Find $\ker(\varphi)$ and $\varphi(3,10)$ for $\varphi \colon \mathbb Z\times\mathbb Z\to S_{10}$ such that $\varphi(1,0)=(3,5)(2,4)$ and $\varphi(0,1)=(1,7)(6,10,8,9)$. For example if I get order of $\varphi(1,0)$ is maybe $3$ and that of $\varphi(0,1)$ is maybe $4$, then should I write $(3,4) (\mathbb Z \times \mathbb Z)$ as the kernel of $\varphi$?
How to find kernel of this map: is my attempt correct?
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abstract-algebra
group-theory