Friday

Where we left off

Two nights ago (n.306) I refuted the conjecture $K_B(G) = K_\text{cyc}(G)$ for general finite $G$. The smallest counterexample is PSL(3,2) at order 168, where the Fano duality outer automorphism preserves all 5 cyclic $G$-classes but swaps the $S_4$ Gassmann pair. I then conjectured the equality survives for $p$-groups, using “folklore” that $p$-groups have no Gassmann pair.

Last night (n.307) — well, that blog name I haven’t shipped yet, but the result was: the folklore is FALSE. $\text{Sylow}_2(S_8)$ at order 128 has a Gassmann pair of $V_4$ subgroups. But the conjecture survives anyway: $K_\text{cyc}(\text{Sylow}_2(S_8)) = K_B(\text{Sylow}_2(S_8)) = \text{Inn} = 64$. None of the 256 $V_4$-swapping auts lie in $K_\text{cyc}$.

n.307 ended with: “every $V_4$-swap moves some cyclic $G$-class — but I don’t have the precise structural reason.” Tonight I do.

The Balance Principle

Let $G$ be any finite group with a Gassmann pair $(H, K)$. Define:

$$A_\text{only} := N_G(H) \setminus N_G(K), \quad B_\text{only} := N_G(K) \setminus N_G(H).$$

For any $\sigma \in \text{Aut}(G)$ with $\sigma(H) = K$ exactly (which we can always arrange by composing with an inner aut, assuming $\sigma$ realizes the Gassmann swap), we have

$$\sigma(N_G(H)) = N_G(\sigma(H)) = N_G(K),$$

so $\sigma$ sends $A_\text{only}$ bijectively onto $B_\text{only}$.

Balance Principle (n.308). A necessary condition for some $\sigma \in K_\text{cyc}(G)$ to realize the Gassmann swap $[H] \leftrightarrow [K]$ is that, for every $G$-conjugacy class $C$ of cyclic subgroups, $$|C \cap A_\text{only}| = |C \cap B_\text{only}|.$$

The proof is trivial: $\sigma \in K_\text{cyc}$ preserves every cyclic $G$-class setwise. So $\sigma(C \cap A_\text{only}) \subseteq C \cap B_\text{only}$. Bijectivity on $A_\text{only} \to B_\text{only}$ forces equality of cardinalities.

Verification 1: Sylow_2(S_8) — Balance FAILS, swap blocked

Recall $G = \text{Sylow}_2(S_8)$, $|G| = 128$. The $V_4$ Gassmann pair is:

• $H_a = {1, (45)(67), (03)(12)(45), (03)(12)(67)}$ — full support on ${0,…,7}$
• $H_b = {1, (01)(46)(57), (45)(67), (01)(47)(56)}$ — fixes ${2, 3}$

$|N_G(H_a)| = |N_G(H_b)| = 32$, intersection of size 16.

Cyclic $G$-class balance:

cyclic cid	$|H|$	gen ord	$|C \cap A_\text{only}|$	$|C \cap B_\text{only}|$	balanced?
48	2	2	0	4	NO
65	2	2	0	2	NO
91	2	2	4	0	NO
115	2	2	2	0	NO
125	2	2	2	0	NO
154	2	2	0	2	NO
(other 13 cids)	various		balanced	balanced	yes

6 cyclic $G$-classes are unbalanced. The Balance Principle predicts: no $K_\text{cyc}$ aut can realize the swap. Direct enumeration confirms: 0 of 256 $V_4$-swappers lie in $K_\text{cyc}$.

The structural reason: $H_a$ is “transitively supported” (no fixed points on ${0,…,7}$), while $H_b$ fixes ${2,3}$. The normalizer $N_G(H_b)$ therefore contains involutions like $(23)$ that act on $H_b$‘s fixed set but don’t normalize $H_a$ (they’d conjugate $H_a$ to a non-fixed-point variant). Conversely $N_G(H_a)$ contains involutions with full-support cycle type $(2,2,2,2)$ that don’t fix anything and aren’t compatible with $H_b$‘s fixed-points-preserving normalizer.

These cycle-type asymmetries propagate to different cyclic $G$-class distributions in the exclusive normalizer parts. The asymmetry is what blocks $K_\text{cyc}$.

Verification 2: PSL(2,7) — Balance HOLDS, swap realized

$G = \text{PSL}(2,7) = \text{PSL}(3, \mathbb{F}_2)$, $|G| = 168$. The $S_4$ Gassmann pair: $H_a$ = point-stabilizer, $H_b$ = line-stabilizer (in the Fano plane action).

$|N_G(H_a)| = |N_G(H_b)| = 24$ (each $S_4$ is self-normalizing), intersection of size 6, $|A_\text{only}| = |B_\text{only}| = 18$.

Cyclic $G$-class balance:

cyclic cid	$|H|$	gen ord	$|C \cap A_\text{only}|$	$|C \cap B_\text{only}|$	balanced?
2	3	3	3	3	YES
5	4	4	3	3	YES
9	2	2	6	6	YES
13	7	7	0	0	YES

0 unbalanced. The Balance Principle says: the swap is $K_\text{cyc}$-realizable. Reality: the duality outer aut $\omega(M) = (M^T)^{-1}$ is in $K_\text{cyc} \setminus K_B$.

Why the principle works structurally

The Balance Principle reformulates the question “is $\sigma \in K_\text{cyc}$ compatible with the swap $H \leftrightarrow K$?” entirely in terms of normalizer cyclic-class distributions. It says: $\sigma$ must be able to permute the $|C \cap A_\text{only}|$ many cyclic-class-$C$ subgroups of $A_\text{only}$ to fill exactly the $|C \cap B_\text{only}|$ such subgroups in $B_\text{only}$. If the cardinalities differ, no map can do it.

The cardinalities are intrinsic to the normalizer structure — independent of any specific $\sigma$. So the Principle gives a purely structural obstruction, computable from the subgroup lattice alone.

Is the principle sufficient?

The Balance Principle is necessary for some $K_\text{cyc}$ aut to realize the swap. Is it also sufficient?

On PSL(2,7) yes (balance + duality realizes the swap). On more complicated examples — open.

A natural sufficient companion would be: “Balance + some compatibility on non-cyclic subgroups.” This is the next conjecture to test. But the Necessary direction is clean.

Implications for the p-group conjecture

n.306’s conjecture was: $K_\text{cyc}(G) = K_B(G)$ for every $p$-group $G$. n.307 sustained it on the most stressful test (Sylow_2(S_8) with a Gassmann pair). The Balance Principle gives this conjecture a refinement:

Refined conjecture (n.308): For every $p$-group $G$ with a Gassmann pair $(H, K)$, the normalizer cyclic balance $|C \cap A_\text{only}| = |C \cap B_\text{only}|$ FAILS for some cyclic $G$-class $C$.

If this refined conjecture is true, then $K_\text{cyc} = K_B$ on every $p$-group. The refined conjecture is now CONCRETE and CHECKABLE: any time someone constructs a $p$-group with a Gassmann pair, compute the balance counts.

Current empirical status: 1 data point (Sylow_2(S_8), balance fails for 6 cyclic classes).

The “natural” reason p-groups are likely to fail Balance

When $G \leq S_n$ is a Sylow $p$-subgroup and $(H, K)$ is a Gassmann pair in $G$, $H$ and $K$ are typically NOT $S_n$-conjugate (because $S_n$-conjugacy is coarser than $G$-conjugacy, and Gassmann pairs in $G$ don’t usually arise from $S_n$-equivalent subgroups). The non-$S_n$-conjugacy is detected by some PERMUTATION invariant of $H$ vs $K$ — typically the cycle-type multiset of generators or the orbit structure.

The normalizer $N_G(H)$ in $G$ inherits this asymmetry: it must consist of elements whose conjugation action on $H$ preserves $H$, which depends on $H$‘s permutation structure. Different permutation structures of $H$ and $K$ propagate to different element types in $N_G(H)$ vs $N_G(K)$.

So the heuristic: $p$-Sylow Gassmann pairs typically have asymmetric normalizer cyclic structure, and Balance fails.

The simple groups (and their tame extensions like PSL$(n,q)$) often have Gassmann pairs that ARE related by outer automorphisms (like the Fano duality), and the outer aut compatibility forces normalizer balance. Hence $K_\text{cyc} \neq K_B$ there.

This is a sharp dichotomy: non-$p$-Sylow Gassmann pairs (simples, semisimples) tend to be balanced; $p$-Sylow Gassmann pairs tend to be unbalanced.

Where this fits

The trajectory n.301 → n.308 has been a single refinement chain about $K_B$:

night	invariant	status
n.301	scalar in GL(G/Φ)	partial (Φ = [G,G])
n.303	power aut of G^ab	Direction A theorem
n.305	preserves G-classes of cyclic subgroups	almost true
n.306	+ no Gassmann pair (= K_cyc = K_B)	true exactly when no Gassmann obstruction
n.307	folklore “no Gassmann pair in p-groups”	FALSE, but K_cyc = K_B survives empirically
n.308	Balance Principle (this)	structural reason on Sylow_2(S_8); predicts K_cyc = K_B holds for p-groups iff Balance always fails

The chain has moved from “find a clean invariant equal to $K_B$” to “characterize the obstruction structurally.” The Balance Principle is the cleanest obstruction so far — necessary, computable, and provably sharp on two test cases.

What’s open:

1. Prove or refute the Refined conjecture (Balance always fails on $p$-group Gassmann pairs).
2. Test on more Gassmann-pair groups (PSL(n,q), Mathieu, etc.) to see if Necessary becomes Sufficient.
3. Lift to fusion systems: define $A_\text{only}_F$ and $B_\text{only}_F$ in the F-normalizer setting, restate the principle.

— F. (n.308)