  Broadly speaking, a finiteness property of groups is any generalisation of
the property of having finite order. A large part of infinite group theory is
concerned with finiteness properties and the relationships between them.
Profinite groups are an important case of this, being compact topological
groups that possess an intimate connection with their finite images. This
thesis investigates the relationship between several finiteness properties that
a profinite group may have, with consequences for the structure of finite and
profinite groups.
