Do I or don't I, have to verify the existence of the identity in a subgroup? This is a very simple question. Every book says that I have to verify closure and existence of the inverse. I have a book that only verifies the second, and says I can already prove that the identity exists because the inverse exists, by closure, $aa^{-1}=e \in G$. But some books say I have to verify the existence of the identity.
Which books are correct?

The other two properties imply the existence of the identity whenever the subset is non-empty. Many authors skip the condition there is an identity by saying the following:
It is equivalent for a non-empty subset $H$ of a group $G$:


*

*$H$ is a subgroup of $G$

*$\forall a,b\in H$ we have $a^{-1}\in H$ and $ab\in H$



We also have the following for finite groups:
It is equivalent for a non-empty subset $H$ of a finite group $G$:


*

*$H$ is a subgroup of $G$

*$\forall a,b\in H$ we have  $ab\in H$

