I was trained as a physicist, rather than a mathematician. So, I apologize if my question is naive.
As a physicist, I think of Lie groups as being continuous groups. I have begun studying the second edition of Brian Hall's book, "Lie Groups, Lie Algebras, and Representations". At the beginning of the book, matrix Lie groups are defined. From the definition in this book, it appears to me that discrete groups can be Lie groups, and all finite groups are Lie groups. This contradicts what I thought Lie groups were.
Matrix Lie groups are defined in Definition 1.4 on p. 4 of the book. This definition is as follows: "A matrix Lie group is a subgroup $G$ of $GL(n;\mathbb{C})$ with the following property: If $A_m$ is any sequence of matrices in $G$, and $A_m$ converges to some matrix $A$, then either $A$ is in $G$ or $A$ is not invertible." The group $GL(n;\mathbb{C})$ is defined to be the general linear group over complex numbers: "the group of all $n \times n$ invertible matrices with complex entries." (I probably did not need to define $GL(n;\mathbb{C})$, but I am not sure since I am not a mathematician.)
From the definitions given above, it appears to me that a matrix representation of a discrete group can be a matrix Lie group. A sequence of matrices in a discrete matrix group should be able to converge to a matrix in the group. Furthermore, any convergent series of matrices in a finite matrix group should only be able to converge to a matrix in the group. Thus, I would conclude that a matrix representation of any finite group is a matrix Lie group. Since all matrix Lie groups are Lie groups, I would conclude that discrete groups can be Lie groups, and all finite groups are Lie groups.
As I explained, I had thought of Lie groups as being continuous. Furthermore, Brian Hall's book defines a Lie group as being a type of smooth manifold. The precise definition is given in Definition 1.20 on p. 25 of the book, and I will not repeat it here. As a physicist, I always thought of manifolds as being continuous objects. I intuitively understand a manifold to be an object whose elements locally can be mapped to $\mathbb{R}^n$. I know that is not a rigorous definition, but that is how I understand manifolds. I do not understand how a discrete group can be a manifold, since I think of manifolds as being continuous. So, I am confused by Brian Hall's definition of matrix Lie groups, which seems to imply that discrete groups can be Lie groups.
Thank you for your help. Once again, sorry if my questions are naive.
