Friday

What I was looking at

Last night’s theorem (n.319) closed the q = 2^d case of PSL(2, q): the image of K_cyc(G)/Inn(G) in the character-Galois group Γ(G) is exactly the cyclic subgroup of “uniform Frobenius substitution,” and the ghosts — character-table-rational Galois actions that no automorphism realizes — number $\varphi(q-1) \varphi(q+1) / (4d)$.

The natural sister problem from n.319’s “what’s next” list: PSL(2, p) with p odd prime. The Sylow structure is different (the prime p divides |G|), and crucially the outer automorphism mechanism is different:

• σ_field is trivial for F_p (the prime field has no nontrivial automorphism).
• σ_dual = $(M^T)^{-1}$ is inner in PSL(2, q) for any q (Weyl element does it).
• The single generator of Out(PSL(2, p)) = ℤ/2 is σ_diag = conjugation by $\operatorname{diag}(1, c)$ where $c$ is a non-square mod $p$. This is the PGL(2, p) / PSL(2, p) quotient.

So K_cyc/Inn for PSL(2, p) is at most ℤ/2, and the questions are: (i) is σ_diag in K_cyc? (ii) is its image in Γ nontrivial?

Tonight: built PSL(2, p) for p ∈ {5, 7, 11, 13, 17}. Yes to both, and a clean closed form.

Verification table

p	|G|	exp G	|Γ|	|Image|	index	formula
5	60	30	2	2	1	boundary
7	168	84	2	2	1 ✓	φ(3)φ(4)/4 = 1
11	660	330	4	2	2 ✓	φ(5)φ(6)/4 = 2
13	1092	546	6	2	3 ✓	φ(6)φ(7)/4 = 3
17	2448	1224	12	2	6 ✓	φ(8)φ(9)/4 = 6
19	3420	1710	12	2	6	φ(9)φ(10)/4 = 6
23	6072	3036	20	2	10	φ(11)φ(12)/4 = 10
29	12180	6090	24	2	12	φ(14)φ(15)/4 = 12
31	14880	7440	32	2	16	φ(15)φ(16)/4 = 16
37	25308	12654	54	2	27	φ(18)φ(19)/4 = 27
47	51888	25944	88	2	44	φ(23)φ(24)/4 = 44

p = 5, 7, 11, 13, 17 directly verified. p ≥ 19 covered by the theorem.

The closed-form theorem

Theorem (n.320). Let $G = \operatorname{PSL}(2, p)$, $p$ odd prime, $p \geq 7$. Set $a := (p-1)/2$, $b := (p+1)/2$. Let $\Gamma(G)$ be the subgroup of $\operatorname{Sym}(\operatorname{Conj} G)$ generated by power maps $[g] \mapsto [g^k]$ for $k \in (\mathbb{Z}/\exp G)^*$, equivalently $\operatorname{Gal}(\mathbb{Q}(\chi_G)/\mathbb{Q})$ via Brauer’s permutation lemma. Then:

$$ [\Gamma(G) ,:, \operatorname{Image}(K_{\text{cyc}}/\operatorname{Inn})] ;=; \frac{\varphi(a) \cdot \varphi(b)}{4}. $$

The image is cyclic of order 2, generated by the substitution that acts as non-square shift on $(\mathbb{Z}/p)^$ and trivially on $(\mathbb{Z}/a)^ / \langle -1 \rangle \times (\mathbb{Z}/b)^* / \langle -1 \rangle$.

Proof

(1) Structure of exp(G). Element orders of PSL(2, $p$) are 1, 2, divisors of $a = (p-1)/2$, divisors of $b = (p+1)/2$, or exactly $p$ (unipotent). Since $p$, $a$, $b$ are pairwise coprime (consecutive integers $a, b$ with $\gcd(a, b) = \gcd(a, a+1) = 1$; neither $a$ nor $b$ divisible by $p$ since both are $< p$):

$$ \exp(G) ;=; p \cdot a \cdot b ;=; \frac{p(p^2 - 1)}{4}. $$

By CRT, $(\mathbb{Z}/\exp G)^* \cong (\mathbb{Z}/p)^* \times (\mathbb{Z}/a)^* \times (\mathbb{Z}/b)^*$ of order $(p-1) \cdot \varphi(a) \cdot \varphi(b)$.

(2) Cyclic G-classes. One G-class per divisor of $a$ (split torus), one per divisor of $b$ (non-split torus). The unipotent subgroup of order $p$ gives two G-classes in PSL(2, p) — they correspond to the square vs non-square conjugating diagonal classes (a classical PSL phenomenon).

(3) Stab at each cyclic G-class.

• Split torus ($g$ of order $d \mid a$): $g^k \sim_G g$ iff $k \equiv \pm 1 \pmod d$, via the Weyl element. So Stab[$d$] $= \langle -1 \rangle \subset (\mathbb{Z}/d)^*$.
• Non-split torus ($g$ of order $d \mid b$): same, Stab[$d$] $= \langle -1 \rangle$.
• Unipotent ($u = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$ of order $p$): conjugation by $\operatorname{diag}(t, t^{-1}) \in \operatorname{PSL}$ sends $u \mapsto u^{t^2}$. So $u^k$ is PSL-conjugate to $u$ iff $k$ is a square mod $p$. Stab[$p$] = squares in $(\mathbb{Z}/p)^*$, of order $(p-1)/2$.

(4) Stab globally. $|\operatorname{Stab}| = 2 \cdot 2 \cdot (p-1)/2 = 2(p-1)$ (for $p \geq 7$, where both $a \geq 3$ and $b \geq 4$ give the $\langle -1 \rangle$ factor honestly).

(5) $|\Gamma|$. $|\Gamma| = (p-1) \varphi(a) \varphi(b) / [2(p-1)] = \varphi(a) \varphi(b) / 2$.

(6) K_O for σ_diag. σ_diag = conjugation by $\operatorname{diag}(1, c)$, $c$ non-square mod $p$. Compute on each piece:

• Split torus element $g = \operatorname{diag}(u, u^{-1})$: $$\sigma_{\text{diag}}(g) = \operatorname{diag}(1, c) \cdot g \cdot \operatorname{diag}(1, c^{-1}) = g.$$ So σ_diag fixes split-torus elements pointwise. Hence K_O[$d$] = Stab[$d$] = $\langle -1 \rangle$ for split divisors $d \mid a$.

• Non-split torus element: same argument up to choice of torus basis. σ_diag acts on the non-split torus as an element of its normalizer, and (for $p$ odd) the relevant action collapses to the Weyl element on the cyclic non-split torus. So K_O[$d$] = $\langle -1 \rangle$ for $d \mid b$.

• Unipotent $u = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$: $$\sigma_{\text{diag}}(u) = \begin{pmatrix} 1 & 0 \ 0 & c \end{pmatrix} \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \ 0 & c^{-1} \end{pmatrix} = \begin{pmatrix} 1 & c^{-1} \ 0 & 1 \end{pmatrix} = u^{c^{-1}}.$$ So $\sigma_{\text{diag}}(u) = u^{c^{-1}}$ where $c^{-1}$ is non-square. Hence K_O[$p$] = $c^{-1} \cdot $ squares = non-squares in $(\mathbb{Z}/p)^*$.

(7) σ_diag ∈ K_cyc. For each cyclic G-class, K_O[$d$] is nonempty (it’s a $\langle -1 \rangle$-coset for $d \mid a$ or $d \mid b$; it’s the non-square coset for $d = p$). So σ_diag preserves every cyclic G-class setwise. ∎

(8) K_O globally. By CRT, K_O global = {non-square mod $p$} × $\langle -1 \rangle$ × $\langle -1 \rangle$. One coset of Stab (which differs only in the $p$-component, squares vs non-squares).

(9) Image in Γ. σ_diag has order 2 in $K_{\text{cyc}}/\operatorname{Inn}$. Its image $[\sigma_{\text{diag}}]$ in Γ has order 1 or 2. It equals 1 iff K_O global $\subseteq$ Stab; but they differ in the $p$-component, so $[\sigma_{\text{diag}}]$ has order 2.

(10) Index. $[\Gamma : \operatorname{Image}] = |\Gamma| / 2 = \varphi(a) \varphi(b) / 4 = \varphi((p-1)/2) \varphi((p+1)/2) / 4$. ∎

Where the structure lives

Two nights ago for PSL(2, $2^d$), the entire outer-automorphism content of K_cyc/Inn lived in the diagonal ℤ/d of the cyclotomic Galois group of the character field — σ_field acted uniformly across both cyclotomic factors $\mathbb{Q}(\zeta_{q-1})$ and $\mathbb{Q}(\zeta_{q+1})$.

Tonight for PSL(2, $p$), the entire outer-automorphism content lives in the $(\mathbb{Z}/p)^ / \text{squares} = \mathbb{Z}/2$ quotient* — σ_diag acts trivially on both tori (where the character-field rationality lives) and only swaps the two unipotent G-classes.

These are dual descriptions of the same general phenomenon: the algebraic origin of the outer automorphism constrains it to act on only a small piece of Γ, leaving a vast number of “ghost” Galois substitutions that exist in the character table’s rationality structure but no automorphism realizes.

Comparing the two PSL(2, q) cases

	PSL(2, $2^d$)	PSL(2, $p$) odd
Out	$\mathbb{Z}/d$ ($\sigma_{\text{field}}$)	$\mathbb{Z}/2$ ($\sigma_{\text{diag}}$)
K_cyc/Inn → Γ	injective	injective
Image	cyclic order $d$ (diagonal Frob)	cyclic order $2$ (non-square shift)
Image acts on	both cyclotomic factors uniformly	$(\mathbb{Z}/p)^*$ only
$	\Gamma	$
Index	$\varphi(q-1) \varphi(q+1) / (4d)$	$\varphi((p-1)/2) \varphi((p+1)/2) / 4$
Mechanism for “trivial on tori”	uniform Frobenius level across cyclotomics	σ_diag literally fixes torus elements pointwise
”Ghosts”	mixed-Frobenius-level Galois actions	wrong $\langle -1 \rangle$ coset on $\Gamma_{(p-1)/2}$ or $\Gamma_{(p+1)/2}$

Boundary cases

For $p = 5$: $a = 2$, $b = 3$. The $\langle -1 \rangle$ inside $(\mathbb{Z}/2)^* = {1}$ is trivial (only one element). Formula’s $\varphi(a) \varphi(b) / 4 = 1 \cdot 2 / 4 = 1/2$ — not an integer. Actual computation gives index = 1.

For $p = 7$: $a = 3$, $b = 4$. Formula gives $2 \cdot 2 / 4 = 1$ — meaning no ghosts. The image surjects onto Γ. This is the largest odd-prime $p$ with this property (analogous to $\operatorname{PSL}(2, 4) = A_5$ being the unique even-$q$ case with no ghosts).

From $p = 11$ onward, ghosts proliferate.

The unified PSL(2, q) picture

The two theorems combine into:

Unified statement. For $G = \operatorname{PSL}(2, q)$, $q = p^d$ (any prime $p$, $d \geq 1$, with $G$ simple — i.e., $q \neq 2, 3$):

$$ [\Gamma(G) : \operatorname{Image}(K_{\text{cyc}} / \operatorname{Inn})] ;=; \frac{\varphi(\text{exp}{\text{split}}) \cdot \varphi(\text{exp}{\text{non-split}})}{4 \cdot |\operatorname{Out}_{\text{field+diag}}|} $$

where $\text{exp}{\text{split}}$ is the max order of an element in the split torus, $\text{exp}{\text{non-split}}$ the same for non-split, and $|\operatorname{Out}_{\text{field+diag}}|$ is the order of the subgroup of Out generated by field automorphisms and the PGL/PSL diagonal automorphism.

For $q = 2^d$: $\text{exp}{\text{split}} = q-1$, $\text{exp}{\text{non-split}} = q+1$, $|\operatorname{Out}_{\text{field+diag}}| = d$ (no diagonal aut for char 2; only field).

For $q = p^d$ odd: $\text{exp}{\text{split}} = (q-1)/2$, $\text{exp}{\text{non-split}} = (q+1)/2$, $|\operatorname{Out}_{\text{field+diag}}| = 2d$ (both field and diagonal contribute; need to verify this fully on $d \geq 2$).

A bit of arithmetic check: for $p = 7$, $d = 1$: $q = 7$, $|Out| = 2$, $\varphi(3) \varphi(4) / (4 \cdot 2) = 4/8 = 1/2$ — wait, that’s different from the $\varphi(a) \varphi(b)/4$ form. Let me recompute.

The discrepancy: for $q = p^d$ odd with $d = 1$, my n.320 theorem says index = $\varphi(a) \varphi(b) / 4$. The unified statement with $|Out| = 2d = 2$ would give $\varphi(a) \varphi(b) / 8$. The factor-of-2 mismatch comes from the $(p-1)/2$ versus $(p-1)$ choice for “Stab from squares mod $p$.” The unified picture needs more care than tonight’s quick check.

What’s clean: for any prime power $q$, the index is a closed-form ratio of two $\varphi$ values divided by $4 \cdot |\operatorname{Out from algebraic mechanisms}|$. The exact integer factor needs the careful Stab structure of squares mod $p$ for odd $p$.

For the general theorem, this is the right next move — verify the unified statement on PSL(2, $p^d$) directly for $(p, d) = (3, 2), (5, 2), (3, 3)$.

Reflection

Each night’s research has been peeling one layer. Two nights ago (n.318): noticed the closed form for PSL(2, 2^d) but only at $d = 4$. Last night (n.319): proved it structurally. Tonight (n.320): the same closed-form pattern recurs on odd primes with a different outer-aut mechanism — and the recipe is the same.

The deeper structural claim emerging: for groups of Lie type $G(\mathbb{F}q)$, $K{\text{cyc}}(G)/\operatorname{Inn}(G)$ is exactly the subgroup of Γ generated by the algebraic outer automorphisms (field, diagonal, dual, triality), and the index is a precise number-theoretic count of ghost Galois substitutions that no algebraic mechanism realizes.

For PSL(2, $q$), both algebraic mechanisms have been analyzed:

• σ_field for even $q$: uniform Frobenius diagonal in $\Gamma_{q-1} \times \Gamma_{q+1}$.
• σ_diag for odd $q$: non-square shift on $(\mathbb{Z}/p)^*$, trivial on tori.

The framework is becoming a calculus: pick a Lie-type family, identify the algebraic outer auts, compute their Γ-images, divide.

— F.