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For a finite p-group in characteristic p, the augmentation ideal of the group algebra is nilpotent

Statement

Let $P$ be a finite $p$ -group and let $k$ be a field of characteristic $p$ . Then the augmentation ideal $I_{P} \subseteq k [P]$ is nilpotent.

Facts & Assumptions

Given: A finite $p$ -group $P$ , a field $k$ of characteristic $p$ , and the augmentation ideal $I_{P} \subseteq k [P]$ .

[F1]

The augmentation ideal is the kernel of the augmentation map (The augmentation map $ε : R [G] \to R$ and the augmentation ideal $I_{G} = \ker ε$ ).

[L1]

The basis elements $[g]$ multiply as $[g] [h] = [g h]$ in $k [P]$ (The group ring $R [G]$ is a unital $R$ -algebra with basis $G$ , and each $g \in G$ is a unit of $R [G]$ ).

[L2]

The algebra $k [P]$ has dimension $∣ P ∣$ (If $G$ is finite then $\dim_{k} k [G] = ∣ G ∣$ ).

[L3]

Every nontrivial finite $p$ -group has a nontrivial central element of order $p$ (Every nontrivial finite $p$ -group has nontrivial center, in fact $p$ divides $∣ Z (P) ∣$ ).

Proof

technique · direct

1.1F1L1L3givenchooseinductionalgebra

We argue by induction on $∣ P ∣$ . If $P = 1$ , then $I_{P} = 0$ by [F1], so the claim is trivial. Assume $P \neq 1$ . By [L3], choose a central element $z \in Z (P)$ of order $p$ . Then $([z] - 1)^{p} = [z^{p}] - 1 = 0$ in characteristic $p$ , using [L1] and the binomial theorem with vanishing intermediate coefficients. Hence the principal ideal generated by $[z] - 1$ is nilpotent.

2.1F1L2step 1.1inductionalgebra∎

Because $z$ is central, the quotient map $P \to P / ⟨ z ⟩$ induces an algebra surjection $k [P] \to k [P / ⟨ z ⟩]$ whose kernel is the ideal $([z] - 1) k [P]$ . Under this map the augmentation ideal of $k [P]$ maps onto the augmentation ideal of $k [P / ⟨ z ⟩]$ . By the induction hypothesis, some power of $I_{P}$ lands inside $([z] - 1) k [P]$ . Since step 1.1 makes that ideal nilpotent, a further power of $I_{P}$ is $0$ . Therefore $I_{P}$ is nilpotent.

Depends on

Used by

Dependency tree · two levels

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Sources

Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)