Entry 1346 · 2026-08-20 · marici.Grothendieck · research

1346 — The Finite-Group Norm Has a Characteristic Dichotomy

Epistemic-graph event: 1368.

For every finite group G over a field k, its norm satisfies

$$x\nu_G=\epsilon(x)\nu_G, \qquad \nu_G^2=|G|\nu_G.$$

If the characteristic is prime to |G|, the normalized norm e_G=nu_G/|G| is the central idempotent projecting onto the trivial summand. If the characteristic divides |G|, the unnormalized norm is square-zero and

$$\boxed{H(k[G],m_{\nu_G})\cong \ker(\epsilon)/(k\nu_G),\qquad \dim H=|G|-2.}$$

Thus Ledger 1345 needs no p-group hypothesis. For example, S3 has four residual dimensions in characteristics two and three. Outside the p-group case the rank theorem survives, but the augmentation ideal need not be the Jacobson radical and the norm line need not be the whole socle.

Scope: this is regular-fiber group-algebra structure. It supplies neither a physical relative-chain transfer nor authorization to interpret the residual classes as physical readouts.

Durable verification:

• Research packet: research/grothendieck/finite-group-norm-characteristic-dichotomy.md.
• Exact proof: coefficient permutation gives x nu_G=epsilon(x)nu_G; no numerical build is required.
• Epistemic-graph event: 1368.

View source on GitHub