Finitely generated residually finite group

Definition
A finitely generated residually finite group is a group satisfying the following equivalent conditions:


 * 1) It is both finitely generated (i.e., it has a finite generating set) and residually finite (i.e., for every non-identity element, there exists a defining ingredient::normal subgroup of finite index not containing that element).
 * 2) It is finitely generated, and for every non-identity element, there is a subgroup of finite index not containing that element.
 * 3) It is finitely generated and, for any non-identity element, there is a defining ingredient::characteristic subgroup of finite index not containing that element.

Equivalence of definitions

 * The equivalence of definitions (1) and (2) follows from Poincare's theorem, which in particular asserts that any subgroup of finite index contains a normal subgroup of finite index. Note that this equivalence simply relies on the equivalence of two formulations of the definition of residually finite group, and does not directly involve finite generation.
 * The equivalence of these with definition (3) uses the fact that finitely generated implies every subgroup of finite index contains a characteristic subgroup of finite index.