Residually finite group
A group is said to be finitely approximable or residually finite if it satisfies the following equivalent conditions:
- For any non-identity element of the group, there is a subgroup of finite index not containing that element.
- 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.
- The intersection of subgroups of finite index in it is trivial.
- The intersection of normal subgroups of finite index in it is trivial.
- The natural map from the group to its profinite completion, is injective.
- 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.
- It is isomorphic to a subgroup of a direct product of finite groups.
- Under the profinite topology, it is a Hausdorff space.
- 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.
- 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 contains a normal subgroup of index dividing , 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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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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In terms of the residually operator
This property is obtained by applying the residually operator to the property: finite group
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Relation with other 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|
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|Subgroup-closed group property||Yes||residual finiteness is subgroup-closed||If is a residually finite group and is a subgroup of , then 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 and a normal subgroup such that the quotient group is not a residually finite group.|
|Direct product-closed group property||Yes||residual finiteness is direct product-closed||If is a collection of residually finite groups, the external direct product of the s is also a residually finite group.|