I am studying Grothendieck group, and I have the following in my mind.
Let $M$ be a monoid and $N$ be a submonoid of $M$. If $\Gamma(M)$ is the Grothendieck group $M$ and $\Gamma(N)$ is the Grothendieck group of $N$, then is it true that $\Gamma(N)$ embeds in $\Gamma(M)$?
I think there should be a homomorphism from $\Gamma(N)$ to $\Gamma(M)$, not an isomorphism.
Yes, you are correct. There is a natural homomorphism from Γ(N) to Γ(M) induced by the inclusion map of N into M, which maps each element of Γ(N) to its equivalence class in Γ(M). However, this homomorphism need not be an isomorphism in general.
To see this, consider the following simple example: Let M be the non-negative integers with addition as the monoid operation, and let N be the even non-negative integers. Then Γ(M) is isomorphic to the integers, while Γ(N) is isomorphic to the integers divided by 2. The natural homomorphism from Γ(N) to Γ(M) sends each even integer n to the integer 2n in Γ(M), which is an injection but not an isomorphism. This shows that in general, the Grothendieck group of a submonoid need not be isomorphic to a subobject of the Grothendieck group of the containing monoid.