# Thread: Residually finite groups problems.

1. ## Residually finite groups problems.

I'm really appreciate if anyone can help me in these proofs.
Urgent homework..

1. Subgroups of residually finite groups are residually finite.
2. Direct products of residually finite groups are residually finite.
3. Finite extensions of residually finite groups are residually finite.

and..
are quotient groups of residually finite groups again residually finite? why? counter example?

2. Originally Posted by deniselim17
I'm really appreciate if anyone can help me in these proofs.
Urgent homework..

1. Subgroups of residually finite groups are residually finite.
let G be residually finite and H a subgroup of G. let $\displaystyle 1 \neq h \in H.$ then there exists a finite group $\displaystyle K$ and a homomorphism $\displaystyle f: G \longrightarrow K$ such that $\displaystyle f(h) \neq 1.$ now consider $\displaystyle \tilde{f}=f|_H.$

2. Direct products of residually finite groups are residually finite.
let $\displaystyle \{G_i: \ i \in I \}$ be a family of residually finite groups and put $\displaystyle G=\bigoplus_{i \in I} G_i.$ let $\displaystyle 1 \neq g=(g_i) \in G.$ if $\displaystyle g_i \neq 1,$ then there exists a finite group $\displaystyle K_i$ and homomorphism $\displaystyle f_i: G_i \longrightarrow K_i$ such that

$\displaystyle f_i(g_i) \neq 1.$ if $\displaystyle g_i=1,$ then define $\displaystyle K_i=\{1\},$ and $\displaystyle f_i(x)=1, \ \forall x \in G_i.$ note that since G is a direct product, $\displaystyle g_i \neq 1$ for only finitely many $\displaystyle i.$ thus $\displaystyle K=\bigoplus_{i \in I} K_i$ is a finite group. define $\displaystyle f=(f_i)_{i \in I}.$

then $\displaystyle f: G \longrightarrow K$ is a homomorphism and $\displaystyle f(g) \neq 1.$

3. Finite extensions of residually finite groups are residually finite.
i'm not sure exactly what you mean here! so we have an exact sequence of groups: $\displaystyle 1 \longrightarrow N \longrightarrow G \longrightarrow L \longrightarrow 1.$ now which one is assumed to be finite and which one residually finite?

are quotient groups of residually finite groups again residually finite? why? counter example?
no! because every group is a quotient of a free group and all free groups are residually finite. so if the above was correct, then every group would be residually finite, which is obviously not true.

3. Originally Posted by NonCommAlg
i'm not sure exactly what you mean here! so we have an exact sequence of groups: $\displaystyle 1 \longrightarrow N \longrightarrow G \longrightarrow L \longrightarrow 1.$ now which one is assumed to be finite and which one residually finite?
sorry for my idiotic. i don't understand $\displaystyle 1 \longrightarrow N \longrightarrow G \longrightarrow L \longrightarrow 1.$.
1 is identity, N is the normal subgroup, G is the group, then what is L?
why it goes to 1 again?

i think the proof should be
" Finite extensions of finitely generated residually finite groups are residually finite."
this makes sense..