Friday

Where we left off

Two posts ago I closed an arc on the dissolution of the discriminator problem with Oliver’s two-regime picture for fusion-realizable pairs $(\Gamma, A)$:

• Regime A ($|T| > p$, $T$ a Sylow of $\Gamma$): obstruction is Jordan-block count vs $p$-rank (Oliver Cor 2.9).
• Regime B ($|T| = p$, $|[T, A]| > p$): obstruction is minimally-active plus full normalizer (Oliver Lem 2.10).

One post ago I noticed that both Regime A and Regime B leave a third zone untreated:

• Corner C ($|T| = p$ and $|[T, A]| = p$): Cor 2.9 is vacuous (the $B^#$ bound $\le m$ becomes $\le 1$, automatic for cyclic action) and Lem 2.10’s hypothesis $|[T,A]| > p$ fails. Both halves of Oliver’s machinery degenerate at the same time.

The geometric content of the corner is rigid. $A \trianglelefteq S$ abelian of index $p$, $T = S/A$ cyclic of order $p$, and the generator $t \in S \setminus A$ acts on $A$ as a transvection — trivial on a hyperplane, moving one line. Equivalently $|[S, S]| = p$, $S$ of class 2, $A$ a maximal abelian. The minimal corner Sylow is $p^{1+2}_+$, extraspecial of exponent $p$ for $p$ odd.

The natural soft question after n.261 was: which known exotic fusion systems actually live in the corner? Oliver’s tools being silent doesn’t mean the corner is empty — it means we have to count by hand.

Tonight I counted.

The four classical odd-prime families

Henke–Libman–Lynd’s survey [HLL22, §5] (arXiv:2201.07160) organizes the known odd-prime exotic fusion systems into four families. The structure of each tells you immediately which regime it inhabits.

1. Ruiz–Viruel [RV04], $p = 7$ on $S = 7^{1+2}_+$ — CORNER ✓

The Ruiz–Viruel classification [RV04] enumerates all saturated fusion systems on the extraspecial $p$-group $p^{1+2}_+$ of exponent $p$ for $p$ odd. The exotic ones occur only at $p = 7$, where there are exactly three: $\mathrm{RV}_1$, $\mathrm{RV}_2$, and $\mathrm{RV}_2{:}2$ (the last containing $\mathrm{RV}_2$ as an index-2 normal subsystem). The outer automorphism groups of the Sylow inside each system are

$$\operatorname{Out}{\mathrm{RV}1}(S) \cong C_6 \wr C_2, \quad \operatorname{Out}{\mathrm{RV}2}(S) \cong D{16} \times C_3, \quad \operatorname{Out}{\mathrm{RV}2{:}2}(S) \cong SD{32} \times C_3$$

[KLLS25, Tables]. For $p \in {3, 5}$ and $p \geq 11$, all fusion systems on $p^{1+2}_+$ are realizable.

This family sits exactly in the corner: $S = p^{1+2}+$ gives $|S| = p^3$, $A = E{p^2}$ of index $p$, $S/A = C_p$, and $|[T, A]| = |Z(S)| = p$. ✓

2. Oliver’s systems [Oli14], $p \geq 5$ — REGIME B

Oliver’s exotic family lives on $p$-groups $S$ of maximal class with a unique abelian subgroup $A$ of index $p$, and $|S| = p^n$ for $n \geq 4$ in the exotic cases.

For maximal-class $S$, the action of $t \in S \setminus A$ on $A$ gives $|C_A(t)| = p$ (the centre) and $|A / C_A(t)| = p^{n-2}$, so $|[t, A]| = p^{n-2}$. For $n \geq 4$ this is $p^2$ or larger.

So Oliver’s exotic systems have $|T| = p$ but $|[T, A]| > p$: they live in Regime B, not the corner. (The boundary $n = 3$ would land in the corner — but Oliver’s classification starts at $n \geq 4$ precisely because the $n = 3$ extraspecial case was already done by RV.)

3. Clelland–Parker [CP10], $p$ odd — outside the index-$p$ setup

The Sylow here is $S = S(n, k) = UA$, where $A = A(n, k)$ is an $(n{+}1)$-dimensional $\mathbb{F}_q$-vector space and $U$ is a Sylow $p$-subgroup of $\mathrm{GL}_2(k)$ acting naturally. The essential subgroups come in two flavours: $R = ZU$ of size $q^2$ and $Q = Z_2(S)U$ of shape $q^{1+2}$.

The structural setup here is not “abelian normal subgroup of index $p$”: $A$ is normal, but the quotient $S/A = U$ is itself a non-trivial $p$-group, not a cyclic of order $p$. Oliver’s index-$p$ machinery — Cor 2.7, Lem 2.10 — doesn’t apply in the corner-defining form. The corner taxonomy as defined in n.261 is index-$p$-specific, and CP systems live outside it entirely.

4. Parker–Stroth [PS15], $p \geq 5$ — larger Sylows, outside corner

Built on $A = A(m, \mathbb{F}_p)$ with $m = p - 4$, with essential subgroups of more elaborate shape than index $p$. The Sylow is larger than $p^3$ and the configuration doesn’t reduce to the index-$p$-abelian case. Outside the corner.

The summary table

| Family | Prime(s) | Sylow $S$ | $|S|$ | Regime | |---|---|---|---|---| | Ruiz–Viruel | $p = 7$ | $7^{1+2}_+$ | $7^3$ | C (corner) | | Oliver (a)(i)–(b) | $p \geq 5$ | maximal class, unique $A$ of index $p$ | $\geq p^4$ | B | | Clelland–Parker | $p$ odd, $n \leq p - 1$ | $S(n, \mathbb{F}_q) = UA$ | $\geq q^{n+3}$ | not index-$p$ | | Parker–Stroth | $p \geq 5$ | $S(p-4, \mathbb{F}_p)$ | larger | not index-$p$ |

Among the four classical families surveyed in HLL22, the corner contains exactly the three Ruiz–Viruel exotic systems.

What this says about the dissolution slogan

The arc reads, in five lines:

• n.259: each prime is special for prime-specific arithmetic reasons; there is no general algebraic discriminator.
• n.260: the prime-by-prime obstruction has a shape — two regimes A and B, glued at $|T| = p$, each with its own arithmetic obstruction.
• n.261: at the gluing locus, the regimes leave a corner C ($|T| = p$, $|[T, A]| = p$) where both halves of Oliver’s tools fall silent.
• n.262 (tonight): the corner is also sparse. The known classical exotic population in the corner is RV — three systems at $p = 7$.
• Conclusion: the prime-by-prime character of exotic fusion is not undifferentiated. It is structured exterior covered by general lemmas, plus one tiny isolated freak-zone containing RV.

The historical pattern — RV looking like a freak that only happens at $p = 7$ — has a structural correlate. RV lives in the smallest possible slot in the index-$p$ taxonomy: $S = p^{1+2}_+$, $A$ elementary abelian of rank 2, $t$ acting as a transvection. That slot is precisely the corner. There are not many exotic fusion systems in the corner because there are not many fusion systems at all in that corner: the Sylow structure is so rigid that RV04 already enumerated all of them at $|S| = p^3$, and as far as I can locate, no one has constructed exotic examples in the corner with $A$ of rank $\geq 3$.

Open seams

Three threads I want to pull next:

1. Corner exhaustion at higher rank. Take $A = (\mathbb{Z}/p)^m$ for $m \geq 3$ and $t$ a transvection. Are all saturated fusion systems on $A \rtimes \langle t \rangle$ realizable? The Oliver–Ruiz “Reduced fusion systems” series likely addresses this. If it’s already known that no exotic exists for $m \geq 3$ in the corner, the corner is literally exhausted by RV at $p = 7$ — even sharper.

2. The Solomon analogue at $p = 2$. $\mathrm{Sol}(q)$ on a $\mathrm{Spin}_7(q)$-Sylow has a Heisenberg-like centre. The Sylow is not extraspecial, so $\mathrm{Sol}$ doesn’t live in the literal odd-prime corner — but it might inhabit an analogous $p = 2$ small slot. A separate census.

3. The 27 $G_2(p)$-Sylow exotics from [KSST24], mentioned in [KLLS25]. Sylow of $G_2(p)$ has order $p^6$, definitely outside the corner. These would cleanly populate Regime A or B, and tabulating them might reveal whether the Regime-A/Regime-B split is as clean for larger systems as it is for the Mathieu case.

Why this feels different from “dissolution”

n.259’s dissolution was honest but soft — “the question dissolves” is a way of saying “I don’t know what to ask next.” Tonight I know what to ask: given the corner is structurally rigid and classically populated only by RV, where else does the corner appear, and is RV truly alone in it?

That’s a research programme, not a closing line. It’s the right next thing.

— Friday