Residually finite group

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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 subgroup of finite index not containing that element.
  2. For any non-identity element of the group, there is a 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 subdirect product of (possibly infinitely many) finite groups, i.e., it is isomorphic to a subgroup of a 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 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.


This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Formalisms

In terms of the residually operator

This property is obtained by applying the residually operator to the property: finite group
View other properties obtained by applying the residually operator

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

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite group order is finite residually finite not implies finite Conjugacy-separable group, Direct product of finite groups, Finitely generated profinite group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, LERF group, Profinite group|FULL LIST, MORE INFO
direct product of finite groups external direct product of (possibly infinitely many) finite groups Profinite group|FULL LIST, MORE INFO
conjugacy-separable group any two distinct conjugacy classes can be separated in a finite quotient conjugacy-separable implies residually finite residually finite not implies conjugacy-separable |FULL LIST, MORE INFO
finitely generated residually finite group residually finite as well as finitely generated residually finite not implies finitely generated |FULL LIST, MORE INFO
finitely generated abelian group finitely generated as well as an abelian group finitely generated abelian implies residually finite (there are finite non-abelian groups) Finitely generated conjugacy-separable group, Finitely generated residually finite group, Finitely presented conjugacy-separable group|FULL LIST, MORE INFO
LERF group every finitely generated subgroup is closed in the profinite topology |FULL LIST, MORE INFO
free group has a freely generating set free implies residually finite (obvious, since nontrivial finite groups are residually finite but not free) |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
locally residually finite group every finitely generated subgroup is residually finite |FULL LIST, MORE INFO
group in which no non-identity element has arbitrarily large roots |FULL LIST, MORE INFO

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
Subgroup-closed group property Yes residual finiteness is subgroup-closed If G is a residually finite group and H is a subgroup of G, then H is also a residually finite group.
Quotient-closed group property No residual finiteness is not quotient-closed It is possible to have a residually finite group G and a normal subgroup N such that the quotient group G/N is not a residually finite group.
Direct product-closed group property Yes residual finiteness is direct product-closed If G_i, i \in I is a collection of residually finite groups, the external direct product of the G_is is also a residually finite group.