Can two non-isomorphic finitely presented groups have isomorphic abelianizations?

Question

I know that if two finitely presented groups, $G$ and $H$, have non-isomorphic abelianizations, then $G$ and $H$ are not isomorphic.

I am wondering if the converse is also true —- if $G$ and $H$ have isomorphic abelianizations, can we conclude that $G$ and $H$ are isomorphic? My intuition tells me this can’t hold in general, but I’m having a hard time coming up with a counterexample.

Answer 1

Yes certainly. As @lulu notes just take a non-abelian group. I.e. one which differs from its abelianization.

After all the abelianization of the abelianization is the abelianization (abelianization is sort of idempotent, similar to closure in topology, or projection in linear algebra; doing it twice is the same as doing it once).

So $S_3$?