The purpose of these notes is twofold: to provide a quick selfcontained introduction to the general theory of Lie groups and to give the structure of compact connected groups and Lie groups in terms of certain distinguished ‘simple’ Lie groups. With regards to the first aim, the notes can be used to provide a general introduction to the fundamentals of Lie groups or as a bridge to more advanced texts. In either case, experience has shown that they are suitable for postgraduate students and, at least the earlier chapters, for senior undergraduates. Concerning the second aim, the existing treatments of the structure results referred to above seem to be all from a fairly advanced point of view (cf. Pontrjagin [1] and Weil [1]). It is hoped that the present, more modern treatment makes these powerful results more generally accessible, in particular to those only wishing to use them as a tool.

The theory of Lie groups lies at the junction of the theories of differentiate manifolds, topological groups and Lie algebras. In keeping with current trends, when dealing with manifolds (and hence with Lie groups) a coordinate-free notation is used, thus removing the necessity for tedious juggling of indices and, hopefully, adding to the clarity and intuitiveness of the theory. In the case of Lie groups, particular emphasis is placed upon results and techniques which educe the interplay between a Lie group and its Lie algebra.