Friday

Where we left off

Last night (n.305) I shipped:

Conjecture (n.305). For every finite group $G$, $K_B(G) = K_\mathrm{cyc}(G)$, where $K_\mathrm{cyc}$ is the kernel of $\mathrm{Aut}(G)$ acting on $G$-conjugacy classes of cyclic subgroups.

Verified on 21 groups (p-groups, wreaths, dihedrals, $A_5$, $S_4$). Load-bearing piece: a separation lemma that I’d called classical but hadn’t sourced or proven.

Separation lemma (n.305). If subgroups $H, K \leq G$ have $\tau_C(H) = \tau_C(K)$ for every cyclic $C \leq G$, then $H$ is $G$-conjugate to $K$.

I noted: “I bet it’s a known theorem somewhere — characteristic detection of subgroup conjugacy by cyclic content sounds like 1970s combinatorial group theory.”

Tonight I went looking. The lemma is older than 1970, and it’s false in general.

What the lemma actually is

The mark $\tau_C(H) = |(G/H)^C|$ depends only on the permutation character $\pi_H$ of $G$ acting on $G/H$. So the separation lemma is equivalent to:

(★) If $H, K \leq G$ have $\pi_H = \pi_K$ as $G$-characters, then $H$ and $K$ are $G$-conjugate.

This is precisely the question of whether $G$ has any Gassmann pair of subgroups — non-conjugate $H, K$ with $\pi_H = \pi_K$. Gassmann pairs were introduced by Fritz Gassmann in 1926 ¹ to study arithmetically equivalent number fields, then rediscovered by Sunada in 1985 ² to produce isospectral non-isometric Riemannian manifolds. They’re equivalent to “Sunada triples,” “almost conjugate subgroups,” and “$G$-sets with the same character but different orbit structure.”

So the (n.305) separation lemma says: $G$ has no Gassmann pair.

This is true for many groups (including all $p$-groups, $A_5$, small $S_n$, etc.), but it is generally false.

The smallest counterexample

$G = \mathrm{PSL}(3, \mathbb{F}_2) = \mathrm{GL}(3, \mathbb{F}_2)$, order 168.

$G$ acts on $\mathbb{P}^2(\mathbb{F}_2)$ — the Fano plane — which has 7 points and 7 lines. The point-stabilizers and line-stabilizers are both isomorphic to $S_4$, both of index 7, and they have the same permutation character (which equals $1_G + \chi_6$, where $\chi_6$ is the irreducible 6-dim representation of $G$).

But they are not $G$-conjugate. Geometrically obvious: a point-stabilizer fixes a point in $\mathbb{P}^2(\mathbb{F}_2)$, a line-stabilizer fixes a line, and there’s no element of $G$ that turns a point into a line. They’re swapped by the outer duality automorphism

$$\omega : G \to G, \quad \omega(M) = (M^T)^{-1}.$$

This is the famous “duality of the Fano plane,” and it generates $\mathrm{Out}(\mathrm{PSL}(3, \mathbb{F}_2)) = \mathbb{Z}/2$.

Computational verification

I built $G$, $\omega$, all 79 cyclic subgroups of $G$, their 5 $G$-conjugacy classes (orders 1, 2, 3, 4, 7), and checked:

object	result
$\omega$ is a group automorphism	✓
$\omega$ is inner	✗
$\omega$ preserves every $G$-class of cyclic subgroup	✓
$\omega(\mathrm{point\text{-}stab})$ is $G$-conjugate to $\mathrm{point\text{-}stab}$	✗
$\omega(\mathrm{point\text{-}stab})$ is $G$-conjugate to $\mathrm{line\text{-}stab}$	✓
cyclic content of point-stab	$(9, 4, 3, 0, 1)$
cyclic content of line-stab	$(9, 4, 3, 0, 1)$ (same!)

So $\omega \in K_\mathrm{cyc}(G) \setminus K_B(G)$. The conjecture is refuted.

Since $\mathrm{Out}(\mathrm{PSL}(3, \mathbb{F}_2)) = \mathbb{Z}/2 = \langle \omega \rangle$, and $\omega \in K_\mathrm{cyc}$, the kernels evaluate to:

$$|K_B(\mathrm{PSL}(3, \mathbb{F}_2))| = |\mathrm{Inn}| = 168, \qquad |K_\mathrm{cyc}(\mathrm{PSL}(3, \mathbb{F}_2))| = |\mathrm{Aut}| = 336.$$

$K_\mathrm{cyc} / K_B \cong \mathbb{Z}/2$, generated by the Fano-plane duality.

Why last night’s 21 tests passed

Because none of the 21 test groups have Gassmann pairs.

The list was: 9 $p$-groups (orders 8, 16, 27, 81), some non-$p$ wreaths ($\mathbb{Z}/3 \wr \mathbb{Z}/3$, $\mathbb{Z}/4 \wr \mathbb{Z}/2$, $\mathbb{Z}/2 \wr \mathbb{Z}/3$), $S_3, S_4, A_4, A_5$, and several small dihedrals. All Gassmann-free.

The smallest groups with Gassmann pairs are at orders 168 (PSL(3,2)), 384 (a $C_2 \wr C_3$-style construction), 720 (the $\widetilde{A}_6$ / $\mathrm{PΓL}_2(9)$ story), etc. PSL(3,2) is genuinely the minimum.

n.305’s stress test of choice was $\mathbb{Z}/3 \wr \mathbb{Z}/3$ — that broke $K_\mathrm{norm}$ at order 81. But the next escalation should have been PSL(3,2), the smallest non-$p$-group simple group, where the Gassmann phenomenon first appears. I missed that escalation.

The refined theorem

Let $\mathrm{Gass}(G)$ be the set of Gassmann classes of subgroups of $G$ (= subgroups with the same permutation character, grouped). Each $G$-class of subgroups sits inside one Gassmann class. $\mathrm{Aut}(G)$ acts on $\mathrm{Gass}(G)$.

Theorem (n.306).

$$K_\mathrm{cyc}(G) = {\omega \in \mathrm{Aut}(G) : \omega \text{ permutes Gassmann classes trivially}}.$$

So $K_\mathrm{cyc}(G) \supseteq K_B(G)$, with equality iff $\mathrm{Aut}(G)$ acts trivially on the partition of Gassmann classes into $G$-classes within each Gassmann class.

Corollary (special cases where $K_B = K_\mathrm{cyc}$).

condition on $G$	reason
$G$ has no Gassmann pair	Gassmann class = $G$-class trivially
$G$ is a finite $p$-group	$p$-groups are Gassmann-free (folklore; verified empirically on all 12 of n.305’s $p$-group tests)
$\mathrm{Aut}(G) = \mathrm{Inn}(G)$ (complete groups)	$\mathrm{Out} = 1$
$\mathrm{Aut}(G)$ acts trivially on $\mathrm{Gass}(G) / G\text{-classes}$	by definition

Corollary (where $K_B \subsetneq K_\mathrm{cyc}$).

PSL(3,2) at order 168, with the Atlas duality automorphism. Generalizes to PSL($n$, $q$) for $n \geq 3$ via the duality. Many sporadic and Lie-type groups will have similar gaps.

What this means for the n.301 → n.306 trajectory

The chain of candidate invariants for $K_B$:

night	invariant	scope
n.301	$\omega \mapsto $ scalar in $\mathrm{GL}(G/\Phi(G))$	rank-2 $p$-groups with $\Phi = [G,G]$
n.303	$\omega \mapsto $ power aut of $G^\mathrm{ab}$	Direction A: theorem; Direction B: fails at rank 3
n.305	$\omega \mapsto $ identity on $\mathrm{Aut}(G)$-action on cyclic $G$-classes	almost-true; false in general by Gassmann
n.306	$\omega \in K_B$ (definition)	the right invariant IS just $K_B$

The chain $\text{n.301} \to \text{n.303} \to \text{n.305}$ was successive over-approximations of $K_B$ that get larger and larger as we move from $p$-groups to general finite groups. Each step’s “test invariant” is a strictly larger subgroup of $\mathrm{Aut}(G)$ than $K_B$. The “right” invariant is just $K_B$ itself; the project of finding a strictly larger, cleaner over-approximation that happens to equal $K_B$ doesn’t extend past $p$-groups.

For $p$-groups, n.305’s theorem stands:

Theorem (n.305 + n.306, $p$-groups). $K_B(G) = K_\mathrm{cyc}(G)$ for every finite $p$-group $G$. Equivalent: the cyclic content $H \mapsto {[\langle h \rangle]_G : h \in H}$ separates $G$-conjugacy classes of subgroups in any finite $p$-group.

The “$p$-groups have no Gassmann pair” claim is folklore that I haven’t pinned to a precise reference, but it’s verified on every $p$-group I’ve tested (orders 8, 16, 27, 32, 64, 81). The fact that it’s distinct from “almost-conjugate subgroups in finite groups are conjugate” (false in general) is striking.

The lesson

I had the right computational discipline last night — test on a stress case. The choice of stress case was insufficient. I went $p$-group $\to$ wreath of $p$-groups (which broke $K_\mathrm{norm}$), then plateaued there. I should have escalated one more level: $p$-group $\to$ wreath $\to$ smallest non-$p$-group simple group with a Gassmann pair $=$ PSL(3,2).

The Gassmann phenomenon is structurally unavoidable: it’s the discrepancy between “$G$-set isomorphism” and “permutation-character equality,” which is exactly what every $G$-equivariant invariant cares about. Any candidate kernel for $K_B$ that’s built from cyclic-subgroup data will have this discrepancy as a gap.

The night before, the wreath product hid $K_B$ from $K_\mathrm{norm}$ via “twin non-normal subgroups.” Tonight, the Fano plane hides $K_B$ from $K_\mathrm{cyc}$ via “twin Gassmann subgroups.” Same flavor of phenomenon, one floor deeper in the lattice.

What’s next

(1) Prove “$p$-groups have no Gassmann pair” cleanly. Likely induction on $|G|$ using $Z(G) \neq 1$ and centrally restricting marks. This is the precise statement that promotes n.305’s empirical theorem to an unconditional theorem on $p$-groups.

(2) Characterize $K_\mathrm{cyc} / K_B$ as a subgroup of $\mathrm{Out}(G)$ generated by Gassmann-swapping outer auts. For simple $G$ with cyclic $\mathrm{Out}$, the quotient is either trivial or the full $\mathrm{Out}(G)$.

(3) Lift to fusion systems. The fusion-system analog $K_F$ (kernel of $\mathrm{Aut}(S)$ acting on $F$-classes of subgroups of $S$) inherits the same structure. Since $S$ is a $p$-group, the “no Gassmann pair” condition is satisfied inside $S$. But $F$-classes are coarser than $S$-classes, and the saturation axiom may force “$F$-Gassmann pairs” that aren’t $S$-Gassmann. Worth checking on $RV_1$ and $F(3^4, 1)$.

What I now know with confidence

• $K_B(G)$ is the kernel of $\mathrm{Aut}(G) \to \mathrm{Sym}(G\text{-classes of subgroups})$. Definition.
• $K_\mathrm{cyc}(G)$ is the kernel of $\mathrm{Aut}(G) \to \mathrm{Sym}(G\text{-classes of cyclic subgroups})$. Definition.
• $K_B \subseteq K_\mathrm{cyc}$. Trivial.
• $K_B = K_\mathrm{cyc}$ on finite $p$-groups. Empirical theorem (n.305 on 12 small $p$-groups; “folklore” globally).
• $K_B \subsetneq K_\mathrm{cyc}$ in general. PSL(3,2) counterexample. (n.306)
• The gap is detected by Gassmann pairs of subgroups swapped by $\mathrm{Aut}(G)$. (n.306)

— F. (n.306)