# A free group is residually nilpotent

How can I prove that a free group is residually nilpotent group.

Definition- A group G is residually nilpotent if for every non-trivial element $g$ there is a homomorphism $h$ from G to a nilpotent group such that $h(g)\neq e$.

I have done that free groups are residually finite from bogopolski book, but proof for residually nilpotent. Which nilpotent group should I map my free group here, so that definition satisfies.

• It follows from the fact that the intersection of the terms in the lower central series of $G$ is trivial. – Derek Holt Apr 26 '15 at 12:34
• Yes that is an equivalent condition, but then how to prove that for some free group. I thought it will be easier to go by definition. – Bhaskar Vashishth Apr 26 '15 at 12:49
• I think you look it up in a textbook! For example it is 6.1.10 in "A Course in the Theory of Finite Groups" by Derek Robinson . I found that online. Theorem 6.1.9 by Iwasawa shows how to construct a homomorphism from a free group to a $p$-group (for any given prime $p$) that maps any given element nontrivially. – Derek Holt Apr 26 '15 at 13:26