Group Theory Isomorphism $ |G|=10$ and $\mathbb{Z}_{10}$ 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 $y$.... 
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It seems like you're confused as to what an isomorphism actually is. Surely two sets of order 10 are isomorphic as sets but as the other answers have said this doesn't necessarily make them isomorphic as groups. For two groups to have a (group) isomorphism you need a bijective (group) homomorphism to exist between them. This is why your projection idea wont work (it isn't enough for a group isomorphism), try it on the dihedral group of degree 5 and the cyclic group of order 10, none of the bijections will be homomorphisms.
