Checking the group axioms

(a) $(\mathbb{Z},+)$: closed (sum of integers is integer), associative, identity $0$, inverse $-n$. Group (abelian).

(b) $(\mathbb{Z},\times)$: closed, associative, identity $1$. But $2$ has no inverse in $\mathbb{Z}$ ($1/2\notin\mathbb{Z}$). Not a group (it is only a monoid).

(c) $(\mathbb{N},+)$: closed, associative, identity $0$. But no inverse: $-1\notin\mathbb{N}$. Not a group.

(d) $(\mathbb{R}^*,\times)$: closed (product of nonzero reals is nonzero), associative, identity $1$, inverse $1/x$. Group (abelian).

(e) $(\{1,-1\},\times)$: closed ($1\cdot1=1$, $1\cdot(-1)=-1$, $(-1)\cdot(-1)=1$), associative, identity $1$, inverse $1$ and $-1$ are their own inverses. Group of order $2$ (cyclic, generated by $-1$).

Conclusion. (a), (d), (e) are groups; (b), (c) are not.