A group is residually finite when every nonidentity survives in some finite quotient: there are a finite group and a homomorphism with .
Every subgroup of a residually finite group is residually finite, because a finite quotient separating an element of the ambient group also separates it after restriction to the subgroup.
An infinite residually finite group is not a simple group. For a nonidentity element, residual finiteness supplies a map to a finite group in which it survives. Its kernel is proper and cannot be trivial, since an infinite group cannot inject into a finite group.

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Residually finite group by Wikipedia Bot 1
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A group \( G \) is called **residually finite** if for every nontrivial element \( g \in G \) (i.e., \( g \neq e \), where \( e \) is the identity element of the group), there exists a finite group \( H \) and a group homomorphism \( \varphi: G \to H \) such that \( \varphi(g) \neq \varphi(e) \).
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