i need to show that subgroups of residually finite groups are residually finite.
i have 2 definitions for residually finite.
definition 1: A group G is called residually finite if for all nontrivial element in G, there exists a normal subgroup with finite index in G such that .
definition 2: A group G is called residually finite if for all nontrivial element x in G, there is a homomorphism from G to a finite group such that .
NonCommAlg proved the above by using definition 2.
now i try to prove by using definition 1. im stuck somewhere and quite confused where to continue. please give me some hints.
here is my proof:
Let be a residually finite group.
Let be a subgroup of and .
By def. 1, there exists a normal subgroup with finite index in such that .
Let be a normal subgroup with finite index in .
Since intersection of subgroups is also a subgroup, i have as a subgroup of .
So, now i have and .
I apply 2nd isomorphism theorem here and obtained and .
I don't know whether I followed the correct path, but i'm stuck here.
I want to prove that . Please give me some hints.
Thank you.