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Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathrm{gr}_k(G)=\Gamma_G(k)/\Gamma_G(k+1)$ and $$\mathrm{gr}_*(G):=\bigoplus_{k\geq 1}\mathrm{gr}_k(G).$$ Then $\mathrm{gr}_*(G)$ has a graded Lie algebra structure induced from the commutator bracket on $G$. The following lemma is quoted from Braids: Introductory Lectures on Braids, Configurations and Their Applications by A Jon Berrick, Frederick R Cohen, Elizabeth Hanbury, Yan-Loi Wong, Jie Wu:

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Since pure braid groups $P_n$ are residually nilpotent, the lemma provides a testing whether certain group homomorphism $f:P_n\to G$ are faithful by testing whether induced morphisms on the level of Lie algebras $\mathrm{gr}_*(f):\mathrm{gr}_*(P_n)\to\mathrm{gr}_*(G)$ are faithful.

So far everything seems clear; but here is an extract from On Injective Homomorphisms For Pure Braid Groups, And Associated Lie Algebras by F. R. Cohen And Stratos Prassidis:

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(some contents are omitted)

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Now my question is: As we mentioned, Lemma 18.2 provides a testing whether certain group homomorphism $f:P_n\to G$ are faithful by testing whether induced morphisms on the level of Lie algebras $\mathrm{gr}_*(f):\mathrm{gr}_*(P_n)\to\mathrm{gr}_*(G)$ are faithful. Why is Theorem 1.1 also related to this?

By the way, for the notation, in Cohen's paper $E^*_0(G)$ is the same as $\mathrm{gr}_*(G)$ defined before.

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