Defining relations for subgroups of finite index of groups with a finite presentation

Publisher Summary This chapter focuses on defining relations for subgroups of finite index of groups with a finite presentation. It presents a problem of finding a set of defining relations for H by considering a group G with a finite presentation, and a subgroup H of G which is generated by a finite set of words in the generators of G and which is known to be of finite index. The chapter highlights that because the subgroup H is defined by a finite set of words and is of finite index, the Todd–Coxeter coset enumeration process must close. Also by Mendelsohn, a set of Schreier–Reidemeister generators for H can be obtained and a rule for determining in which coset of H a word in G lies. With this information, a set of defining relations for H can be written. Also by Benson and Mendelsohn, the Schreier–Reidemeister generators can be expressed as words in the originally given generators of H .