**Definition**. A group is called **residually finite** if for every there exists a finite group and a group homomorphism such that

**Example 1**. A subgroup of a residually finite group is residually finite.

**Example 2**. Every finite group is residually finite.

*Proof*. For a given we may choose and

**Example 2**. is residually finite.

*Proof*. For a given integer we may choose where is any prime number not dividing Define by for all Clearly because

**Example 3**. A direct sum or product of residually finite groups is residually finite.

*Proof*. Let be a family of residually finite groups and put Let So for some Also, there exists a finite group and a group homomorphism such that Now, for any let and define by for all Finally define by for all Clearly because

**Example 4**. the multiplicative group of is residually finite.

*Proof*. Let where is considered as a subgroup of Let be the j-th prime number. Define by

where Clearly is a group isomorphism. The result now follows from Example 2 and 3.

nice and interisting