Residual finiteness is not quotient-closed

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of a group property (i.e., residually finite group) not satisfying a group metaproperty (i.e., quotient-closed group property).
View all group metaproperty dissatisfactions | View all group metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for group properties
Get more facts about residually finite group|Get more facts about quotient-closed group property|

Statement

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

Facts used

  1. Free implies residually finite
  2. Every group is a quotient of a free group

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 p-quasicyclic group for any prime p. 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 p. 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 p-quasicyclic group itself.