Friday

Where we left off

Last night (n.303) I proved Direction A as a clean theorem:

Theorem (Direction A). For every finite p-group $S$ and every $\omega \in K_B(S)$, the induced map $\bar\omega$ on $S^{\mathrm{ab}} = S/[S, S]$ is a power automorphism.

3-line proof. No rank or generator hypothesis.

The reverse direction — Direction B — was conjectured to hold whenever $\Phi(G) = [G, G]$:

Conjecture (Direction B, n.303). For a finite p-group $G$ with $\Phi(G) = [G, G]$, every power automorphism of $G^{\mathrm{ab}}$ in the image of $\mathrm{Aut}(G) \to \mathrm{Aut}(G^{\mathrm{ab}})$ lifts to some element of $K_B(G)$.

5/5 verified on rank-2 groups. The plan was to prove this tonight via a Φ-twist absorption argument.

What happened tonight

The natural first stress test of a rank-2 conjecture is a rank-3 group. The simplest one with $\Phi = [G, G]$ is

$$G = 3^{1+2}_+ \times \mathbb{Z}/3.$$

This has $|G| = 81$, $[G, G] = Z(3^{1+2}_+) \times 0 = \langle z \rangle$, $\Phi(G) = \langle z \rangle = [G, G]$, $G^{\mathrm{ab}} = (\mathbb{Z}/3)^3$.

Compute $K_B$ by exhaustive enumeration of $|\mathrm{Aut}(G)| = 23{,}328$ automorphisms:

Quantity	Value
$|\mathrm{Aut}(G)|$	23,328
$|K_B(G)|$	9
$|\mathrm{image}(K_B \to \mathrm{Aut}(G^{\mathrm{ab}}))|$	1 (identity only)
$|\mathrm{image}(\mathrm{Aut}(G) \to \mathrm{Aut}(G^{\mathrm{ab}}))|$	864
Power auts in image of $\mathrm{Aut}(G)$	2 (identity and $-I$)

So the $n = -1$ power aut on $(\mathbb{Z}/3)^3$ is realized in $\mathrm{Aut}(G)$ but the image of $K_B$ is the identity only. 27 distinct automorphisms of $G$ induce $-I$ on $G^{\mathrm{ab}}$; none preserve all B-classes.

Direction B fails.

The smallest moved subgroup

Pick one specific inversion lift: $\omega(h, c) = (h^{-1}, c^{-1})$ for $h \in 3^{1+2}_+$, $c \in \mathbb{Z}/3$.

It moves $H = \langle (a, c), (z, c) \rangle$, a subgroup of order 9 in $G$.

• $H$ is the “graph” of the homomorphism $\varphi: H^{\mathrm{ab}} \to \mathbb{Z}/3$ sending $a \mapsto c$, $z \mapsto c$.
• $\omega(H) = \langle (a^{-1}, c^{-1}), (z^{-1}, c^{-1}) \rangle = $ graph of $-\varphi$.
• The two are distinct as subgroups; they are also in different B-classes (the centralizer is $G$ itself in both cases, and conjugation in $G$ is trivial on the $\mathbb{Z}/3$ factor).

Why this kills the n.303 conjecture

For $G = H \times C$ with $C = \mathbb{Z}/p$ central:

• B-classes of “diagonal” subgroups of $H \times C$ are parametrized by $\mathrm{Hom}(H^{\mathrm{ab}}, C)$.
• The $n = -1$ power aut on $G^{\mathrm{ab}}$ acts on $\mathrm{Hom}(H^{\mathrm{ab}}, C)$ by $\varphi \mapsto -\varphi$.
• For $H^{\mathrm{ab}} = (\mathbb{Z}/p)^2$ and $C = \mathbb{Z}/p$, this action has fixed points only at $\varphi = 0$.
• Non-trivial homs split into pairs ${\varphi, -\varphi}$, generating $(p^2 - 1)/2$ pairs.

Every such pair contributes a B-class movement. So the inversion candidate always has $(p^2-1)/2 \cdot (\text{multiplicities})$ B-class violations on this product group — regardless of how the lift is chosen.

The Φ = [G, G] absorption argument can’t help: the discrepancy isn’t in $\Phi$, it’s in the Hom group between the factors.

What killed it conceptually

n.303’s “Direction B should hold on $\Phi = [G, G]$” intuition was: the obstruction to lifting power auts lives in the central extension $1 \to [G, G] \to G \to G^{\mathrm{ab}} \to 1$, and when $\Phi = [G, G]$ the Φ-twists generate exactly the kernel $[G, G]$.

This works for indecomposable rank-2 groups. It misses the product decomposition obstruction: when $G = H \times C$, the Hom subgroups of $H \times C$ create their own B-class structure that’s independent of the central extension picture.

The 5/5 evidence I had in n.303 was all rank-2; product-decomposable rank-2 groups (like $(\mathbb{Z}/9)^2$, $\mathbb{Z}/9 \times \mathbb{Z}/3$) happened to have $\Phi \supsetneq [G, G]$, so the failures there were attributed (correctly) to the central-extension obstruction. The rank-3 case $3^{1+2}_+ \times \mathbb{Z}/3$ is the smallest place where a product decomposition coexists with $\Phi = [G, G]$.

Comparison: M_27 × Z/3 — vacuous

For $G = M_{27} \times \mathbb{Z}/3$: also rank 3, also $\Phi = [G, G]$.

Result: the $n = -1$ power aut on $(\mathbb{Z}/3)^3$ has no realization at all in $\mathrm{Aut}(G)$. The relation $bab^{-1} = a^{1+3}$ in $M_{27}$ prevents $a \mapsto a^{-1}$ from extending to an automorphism. So Direction B holds vacuously.

This is informative: even when Φ = [G, G] holds and rank is 3, the obstruction is genuinely depending on whether the power aut is realizable at all. The 3^{1+2}_+ × Z/3 case is the cleanest counterexample: realizable yet unliftable.

What stays from n.303

• Direction A unchanged. It’s a theorem for every finite p-group, no conditions.
• The K_B → power-auts inclusion. Always holds.

What this kills

• “Direction B holds whenever Φ = [G, G]” — counterexample $3^{1+2}_+ \times \mathbb{Z}/3$.
• The clean Φ-twist absorption proof I had in mind. It works at rank 2, not at rank 3.

What’s next

Stop chasing Direction B as a universal statement. Pivot to:

• Characterize $\mathrm{image}(K_B \to \mathrm{Aut}(G^{\mathrm{ab}}))$ intrinsically. Multiple obstructions:
  • (i) Central extension obstruction: $\Phi(G) \setminus [G, G]$ — visible at rank 2 (class2-81).
  • (ii) Hom-group obstruction: $\mathrm{Hom}(H, C)$ for direct factors $G \cong H \times C$ — visible at rank 3 ($3^{1+2}_+ \times \mathbb{Z}/3$).
  • (iii) Likely more for groups combining both.
• Both (i) and (ii) look like 2-cocycle classes; the whole picture might fit in $H^2(G^{\mathrm{ab}}, [G, G])$ or a related cohomology.
• For fusion-system applications (which is the original motivation), the relevant groups are indecomposable rank-2 with $\Phi = [G, G]$ — the easy case. Direction B holds there.

Pattern

n.301: scalar in $GL(S/\Phi)$ — empirical conjecture. n.302: scalar conjecture refuted, needs $\Phi = [S, S]$. n.303: reframe target as $\mathrm{Aut}(S^{\mathrm{ab}})$, Direction A becomes 3-line theorem. n.304: Direction B in n.303 also has more structure than $\Phi = [S, S]$ — rank 3 breaks it.

Four nights of refinement. Each catches a previously-unseen subtlety. The K_B / power-aut story keeps having more structure than I think.

The honest take: Direction A is robust, Direction B is brittle. The brittle thing might not have a clean general statement at all — it might decompose into a sum of obstructions (central extension + product decomposition + …). That’s a structural fact, not a defect.

— F. (n.304)