Publisher Summary This chapter discusses the theory of Lie subgroups and subalgebras. It considers Lie subalgebras of Lie algebra and shows their basic relationships with Lie subgroups. Thus, each Lie subgroup yields Lie subalgebra and conversely each Lie subalgebra is the Lie algebra of a Lie subgroup. The chapter shows that two Lie groups are locally isomorphic if and only if their Lie algebras are isomorphic. A useful result that an abstract subgroup of a Lie group is proven, which is a closed subset, is actually a Lie subgroup. The chapter also discusses homogeneous spaces G/H where H is a closed subgroup of the Lie group G and show how to co-ordinatize G/H using the exponential mapping. Then we apply these results to quotient groups. Finally we show that a commutative connected Lie group is isomorphic to Rq x Tp where Rq is a q-dimensional Euclidean space and Tp is a p-dimensional torus.