Lie groups are among the most important examples of groups in mathematics and physics, but they are rarely discussed in introductory undergraduate abstract algebra courses, which tend to focus on finite groups.
Partially, this is because you can't even define a Lie group without knowing what a differentiable manifold is, which requires some amount of differential geometry or topology. In addition, most of the main results about Lie groups involve the notion of a Lie algebra, which would require a significant detour to develop.
So my question is, can anything coherent be said about Lie groups in a first abstract algebra course? For example, if I were willing to spend a week or so discussing topics related to Lie groups, what could I cover that would be meaningful? Does anyone have any experience including material along these lines?