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?

Your instinct is correct: if you know the subset is closed and inverses exist, then you know $aa^{-1}  = e \in H \leqslant G $. This is verifying that the identity is present in $H$.
