Residually finite group

Definition
A group is said to be finitely approximable or residually finite if it satisfies the following equivalent conditions:


 * 1) For any non-identity element of  the group, there is a defining ingredient::subgroup of finite index not containing that element.
 * 2) For any non-identity element of the group, there is a defining ingredient::normal subgroup of finite index not containing that element. In other words, there is a homomorphism from the whole group to a finite group where the image of the given element is a non-identity element of the finite group.
 * 3) The intersection of subgroups of finite index in it is trivial.
 * 4) The intersection of normal subgroups of finite index in it is trivial.
 * 5) The natural map from the group to its profinite completion, is injective.
 * 6) It is a defining ingredient::subdirect product of (possibly infinitely many) finite groups, i.e., it is isomorphic to a subgroup of a defining ingredient::direct product of finite groups that surjects onto each of the direct factors.
 * 7) It is isomorphic to a subgroup of a direct product of finite groups.
 * 8) Under the defining ingredient::profinite topology, it is a Hausdorff space.
 * 9) For any finite subset of the group that does not contain the identity element, there is a subgroup of finite index in the group that does not intersect the subset.
 * 10) For any finite subset of the group that does not contain the identity element, there is a normal subgroup of finite index in the group that does not intersect the subset.

Equivalence of definitions
Definitions (1) and (3) are clearly equivalent, as are definitions (2) and (4). The equivalence of definitions (1) and (2) follows from Poincare's theorem which states that any subgroup of index $$n$$ contains a normal subgroup of index dividing $$n!$$, namely its normal core. In particular, any subgroup of finite index contains a normal subgroup of finite index.

Definitions (4) and (5) are equivalent because the kernel of the natural map to the profinite completion is the intersection of normal subgroups of finite index.

Formalisms
The group property of being residually finite is obtained by applying the residually operator to the group property of being finite.