Exercise 27, § 4

Let $A$ be an associative algebra with unit element 1 and let $A_0 = A \supset A_1 \supset \cdots \supset A_n \supset \cdots$ be a decreasing sequence of two-sided ideals of $A$ such that $A_i, A_j \subset A_{i+j}$. Let $G$ be a group with identity element $e$ and let $f : G \to A$ be a mapping such that $f(e) = 1,\ f(xy) = f(x) f(y)$ and $1 - f(x) \in A_1$ for all $x \in G$. Let $H_n$ denote the set of $x \in G$ such that $1 - f(x) \in A_n$. Show that the $H_n$ satisfy the conditions of Exercise 26. Show that the mapping $x \mapsto f(x) - 1$ defines on taking quotients an injective homomorphism of the Lie algebra $\operatorname{gr}(G)$ into the Lie algebra associated with the graded ring $\operatorname{gr}(A) = \sum A_n / A_{n+1}$ (cf. Commutative Algebra, Chapter III).