I think I just didn't get the core of group theory. Although it makes sense to me to follow the regular steps to solve problems of group theory. For example, a group of order $10$ is isomorphic to $\mathbb{Z}_{10}$. To prove this, the standard solution suggests that we have to suppose that there are $2$ elements $x,y$. $x$ has order of 5 another has order of $2$ to begin with. And finally applied that it's isomorphic to $\mathbb{Z}_{2}\times\mathbb{Z}_{5}$ and then isomorphic to $\mathbb{Z}_{10}$
I am confused that, isn't that obvious that a group of order 10 is isomorphic to Z10?? Both of them have $10$ elements. We can simply project them one-by-one.. like $1$ to $x$ ; $2$ to $e$ ; $3$ to $y$.... ....... .......