Definition. A group G is called residually finite if for every 1 \neq g \in G there exists a finite group H and a group homomorphism \varphi : G \longrightarrow H such that \varphi(g) \neq 1.

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

Example 2. Every finite group is residually finite.

Proof. For a given g \in G we may choose H=G and \varphi = \text{id}_G.

Example 2. \mathbb{Z} is residually finite.

Proof. For a given integer n we may choose H = \mathbb{Z}/p \mathbb{Z}, where p is any prime number not dividing n. Define \varphi : \mathbb{Z} \longrightarrow H by \varphi(m)=m+p\mathbb{Z}, for all m \in \mathbb{Z}. Clearly \varphi(n) = n + p \mathbb{Z} \neq 0 because p \nmid n.

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

Proof. Let \{G_i \}_{i \in I} be a family of residually finite groups and put G= \oplus_{i \in I} G_i. Let 1 \neq g = (g_i) \in G. So g_k \neq 1 for some k \in I. Also, there exists a finite group H_k and a group homomorphism \varphi : G_k \longrightarrow H_k such that \varphi_k(g_k) \neq 1. Now, for any k \neq i \in I let H_i = \{1\} and define \varphi_i : G_i \longrightarrow H_i by \varphi_i(x_i)=1, for all x_i \in G_i. Finally define \varphi : G \longrightarrow \bigoplus_{i \in I} H_i by \varphi (x_i) = (\varphi_i(x_i)), for all x = (x_i) \in G. Clearly \varphi(g) \neq 1 because \varphi_k(g_k) \neq 1.

Example 4. \mathbb{Q}^{\times}, the multiplicative group of \mathbb{Q}, is residually finite.

Proof. Let G=\{1,-1\} \oplus \mathbb{Z} \oplus \mathbb{Z} \oplus ..., where \{1,-1\} is considered as a subgroup of \mathbb{Q}^{\times}. Let p_j be the j-th prime number. Define f : \mathbb{Q}^{\times} \longrightarrow G by

f(x)=(\text{sgn}(x),n_1,n_2, \cdots , n_k, 0, 0, \cdots),

where x=\pm p_1^{n_1}p_2^{n_2} \cdots p_k^{n_k}. Clearly f is a group isomorphism. The result now follows from Example 2 and 3.

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Comments
  1. andy says:

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