Exercise 6, § 4

Let $K[G]$ be the algebra of $G$ over $K$ and let $I$ be the kernel of the canonical homomorphism $K[G] \to K$ ("augmentation ideal"). Then

K[G] = K ⊕ I and I admits as basis the family of $g - 1$ where $g \in G - \{e\}$.

(a) If $n$ is an integer $\geqslant 0$, we denote by $I^n$ the ideal of $K[G]$ the $n$-th power of $I$. Let $G_n$ be the set of $g \in G$ such that $g - 1 \in I^n$. Show that $(G_n)$ is an integral central filtration on $G$. In particular, $G_n \supset C^n G$ for all $n$.

(b) Show that if $K = \mathbf{Z}$ the mapping $g \mapsto g - 1$ defines on taking quotients an isomorphism of $G/(G, G)$ onto $I/I^2$. Deduce that $G_2 = C^2 G$.†

(c) Suppose that $K$ is a field of characteristic zero and that $G$ is finite. Show that $I^n = I$ for all $n \geqslant 1$.

(d) Suppose that $K$ is a field of characteristic $p > 0$ and that $G$ is a $p$-group. Show that $I^n\{0\}$ for $n$ sufficiently large. (Show first, using Algebra, Chapter I, § 6, no. 5, Proposition 11, that every simple $K[G]$-module is isomorphic to $K$; deduce that $I$ is the radical of $K[G]$ and hence is nilpotent since $K[G]$ is of finite rank over $K$.)

(e) Suppose that $K = \mathbf{Z}$ and that $G$ has the following property: for all $g \in G$ such that $g \neq 1$, there exists a prime number $p$, a $p$-group $P$ and a homomorphism $f : G \to P$ such that $f(g) \neq e$. Show that in that case $\bigcap_n I^n = \{0\}$. (Reduce it immediately to the case where $G = P$. By applying (d) to the field $\mathbf{F}_p$, we see that there exists $m$ such that $I^m \subset p \cdot \mathbf{Z}[G]$ and, as $I$ is a direct factor in $\mathbf{Z}[G]$, this implies that $I^m \subset pI$, whence $\bigcap_n I^{mn} \subset \bigcap_n p^n I$, which reduces to 0 since $I$ is a finitely generated Abelian group.)