Let $X \subset \Bbb R^m$ and $f : X \rightarrow \Bbb R^n$. We say that $f \in C^k(X, \Bbb R^n)$, $k \geq 1$, if there exists an open set $U \subset \Bbb R^m$ such that $X \subset U$ and a function $\tilde{f} : U \rightarrow \Bbb R^n$ of class $C^k$ in the usual sense such that $\tilde{f}{\restriction_{X}} = f$.
Let $G \subset \Bbb R^n$ be such that $(G, \cdot)$ is a group where $\cdot$ and the inversion are $C^k$-differentiable in the above sense. Is $G$ necessarily a Lie Group (i.e. does there always exist a differentiable structure on $G$ which is compatible with the above definition of differentiability) ?
No. The addition map on $\mathbb{Q}$ sitting inside $\mathbb{R}$ is the restriction of a differentiable function on $\mathbb{R} \times \mathbb{R}$, and similarly subtraction is the restriction of a differentiable function on $\mathbb{R}$, but $\mathbb{Q}$ is not a topological manifold.