Group as a category

Yes, you can. Define a category $M$ with just one formal object say $ob(M) = \{X\}$. Let $G$ be a group. Define $Mor(X,X)$ = underlying set of $G$, and composition of morphisms in $Mor(X,X)$ by the binary operation on $G$. The identity morphism on X is just the identity element in $G$. Then you can verify that all axioms of a category are satisfied by $M$. Since each element in $G$ has an inverse, note, moreover, that every element in $Mor(X,X)$ is an isomorphism. In fact, you can define any monoid as a category with just one formal object in this way. But, with a monoid viewed as a category with one object, it is no longer true that every morphism of that object is an isomorphism.