Residually finite group

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In the mathematical field of group theory, a group G is residually finite or finitely approximable if for every element g that is not the identity in G there is a homomorphism h from G to a finite group, such that

${\displaystyle h(g)\neq 1.\,}$^[1]

There are a number of equivalent definitions:

• A group is residually finite if for each non-identity element in the group, there is a normal subgroup of finite index not containing that element.
• A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial.
• A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial.
• A group is residually finite if and only if it can be embedded inside the direct product of a family of finite groups.