Is the free group of rank 2 a subgroup of an infinite product of finite groups?

Question

I want to find an example of a non-amenable infinite product on amenable groups. My idea is to show that the free group of rank 2 is a subgroup of an infinite product of finite groups.

But I'm not really sure if it is true, and if it is, I don't know how to prove it.

Answer 1

A group is a subgroup of an infinite product of finite groups if (and only if) it is residually finite (i.e., no element is in the kernel of every homomorphism to a finite quotient group), as then it embeds in the product of all of its finite quotients. And free groups are residually finite.