Except in Sects. 4.4 and 4.8, \((W,S)\) is a Coxeter system. A special subgroup of W means one which is generated by a subset of S. In Sect. 4.1 we show, among other things, that special subgroups are Coxeter groups (Theorem 4.1.6). In Sect. 4.3 we show there is a unique element of minimum length in each coset of a special subgroup. This allows us to discuss, in Sect. 4.5, certain “convex subsets” of W such as “half-spaces” and “sectors.” In Sect. 4.6 we prove that each finite Coxeter group has a unique element of longest length. We use this in proving an important result, Lemma 4.7.2: for any given \(w\in W\) , the set of letters with which a reduced expression for w can end generates a finite special subgroup. In Sect. 4.9 we prove that any subgroup of W which is generated by reflections is itself a Coxeter group. In Sect. 4.10 we state a theorem of Deodhar describing normalizers of special subgroups.