# Residual finiteness is not quotient-closed

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., residually finite group)notsatisfying a group metaproperty (i.e., quotient-closed group property).

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## Statement

A quotient group of a residually finite group need not be a residually finite group.

## Facts used

## Proof

### A generic example

Any free group is a residually finite group (fact (1)). Every group is isomorphic to a quotient group of a free group (fact (2)). However, there do exist groups that are not residually finite -- for instance, infinite simple groups.

### A specific example

Consider the -quasicyclic group for any prime . This is a quotient of a direct product of finite groups -- namely, the direct product of all cyclic groups of prime power order for the prime . The latter is clearly residually finite. On the other hand, the quasicyclic group is not -- any quotient by a proper subgroup is isomorphic to the -quasicyclic group itself.