This paper, a continuation of the author's course at McGill, aims to prove that the Galois groups associated with points of finite order on elliptic curves are "as large as possible." The methods used in the course, which focus on points of order \( l^n \) where \( l \) is a fixed prime, become insufficient when \( l \) varies. The main result is that the index of the Galois group \( \varphi_n(G) \) in \( \text{Aut}(E_n) \cong \text{GL}_2(\mathbf{Z}/n\mathbf{Z}) \) is bounded by a constant depending only on \( E \) and \( K \). This is reformulated by considering the limit as \( n \to \infty \), leading to the assertion that \( \varphi_{\infty}(G) \) is a finite index subgroup of \( \text{Aut}(E_{\infty}) \). Further, it is shown that for almost all primes \( l \), \( \varphi_{l}(G) = \operatorname{Aut}(E_{l}) \). The paper builds on previous work by André Weil, particularly the result that \( \varphi_{l \infty}(G) \) is an open subgroup of \( \operatorname{Aut}(E_{l \infty}) \).This paper, a continuation of the author's course at McGill, aims to prove that the Galois groups associated with points of finite order on elliptic curves are "as large as possible." The methods used in the course, which focus on points of order \( l^n \) where \( l \) is a fixed prime, become insufficient when \( l \) varies. The main result is that the index of the Galois group \( \varphi_n(G) \) in \( \text{Aut}(E_n) \cong \text{GL}_2(\mathbf{Z}/n\mathbf{Z}) \) is bounded by a constant depending only on \( E \) and \( K \). This is reformulated by considering the limit as \( n \to \infty \), leading to the assertion that \( \varphi_{\infty}(G) \) is a finite index subgroup of \( \text{Aut}(E_{\infty}) \). Further, it is shown that for almost all primes \( l \), \( \varphi_{l}(G) = \operatorname{Aut}(E_{l}) \). The paper builds on previous work by André Weil, particularly the result that \( \varphi_{l \infty}(G) \) is an open subgroup of \( \operatorname{Aut}(E_{l \infty}) \).