MonsterRoadmap

Group theory roadmap for monster (may be out of date)

1. Multiply transitive actions (Antoine Chambert-Loir has developed a substantial library in his "Jordan" Github repository)

import Mathlib.GroupTheory.GroupAction.Defs -- has class MulAction.IsPretransitive where exists_smul_eq (type may be empty)
import Mathlib.GroupTheory.GroupAction.Basic -- orbit relations
import Mathlib.GroupTheory.GroupAction.FixingSubgroup -- in the presence of `MulAction M α` (with `Group M`) it is the `Subgroup M` consisting of elements which fix `s : Set α` pointwise.
import Mathlib.GroupTheory.Subgroup.Actions -- get action of subgroup, and faithful if original action is faithful.

variable (G : Type u) {α : Type v} [Group G] [MulAction G α]

Define action of FixingSubgroup x on the complement.

class MulAction.IsMultiplyTransitive 
Any action is 0-transitive.
An action on \alpha is (n+1)-transitive if it is transitive (in particular Nonempty \alpha) and for any x : \alpha, the action of FixingSubgroup x on the complement x^c is n-transitive.

Def: Let G act on a set S.  A set of blocks is a partition of S that is preserved by G.  That is, a G-invariant equivalence relation.

Def: An action of G on S is primitive if there is no nontrivial set of blocks (i.e., other than the single block S and the partition into singletons).

Theorem: An action is primitive iff it is transitive and the stabilizer of an element is maximal.

Proof: Given a transitive action, we get a bijection between elements and cosets.  Any intermediate subgroup yields nontrivial blocks via coset decomposition.
Conversely, any nontrivial block containing the element has stabilizer given by an intermediate subgroup.

Lemma: Any 2-transitive action is primitive.

Proof: If 2 elements are equivalent, then all elements are equivalent.

Lemma: Let G act transitively on a set S. If H is normal in G, then the transitivity classes of the action, restricted to H, form a set of blocks for the action of G.

Proof: For two H-orbits T U, transitivity gives us g that sends an element of T to U, and normality makes this a bijection from T to U.

Lemma: Let G act primitively on a set S.  If H is normal in G, then either H is transitive of H is in the kernel of the action.

Theorem: If G acts primitively and faithfully on S, and the stabilizer of a point x is a simple group K, then any proper normal subgroup H is either trivial or acts simply
    transitively on S, and G is a semidirect product of H with K.

Proof: H \cap K is normal in K, so is either trivial or K.
We claim it is not K.  If it were, K \subseteq H, but transitivity means K is maximal, so H = K.  Since H then stabilizes x and is normal, then H stabilizes all points, so is trivial by faithfulness.
Now we have trivial intersection, so no non-identity element of H fixes x, and H acts freely and transitively on S.  Then, G is generated by H and K, since for any g, there is some h such that
hgx = x, meaning hg \in K.  This makes G a semidirect product.

Theorem: If G acts faithfully on S and the stabilizer K of a point x is simple, then the following hold:
1. If G is doubly transitive on S, and H is nontrivial proper normal, then H is elementary abelian, and K is a subgroup of GL_n(F_p), and G is a subgroup of AGL_n(F_p).
2. If G is triply transitive on S, and |S| > 3, then any nontrivial proper normal H is an elementary 2-group.
3. If G is quadruply transitive on S, and |S| > 4, then G is simple.

Proof: From the previous theorem, H is identified with S, and the doubly-transitive property means the conjugation action of K is transitive on H \ {1}.  Thus, H is elementary.
For part 2, if p > 2, then for any x \mapsto y, we must have 2x \mapsto 2y, contradicting double-transitivity on H\{0}. (note: if |S| = 3, we have G = S_3, K = S_2, H = A_3).
For part 3, GL_n(F_2) is not triply transitive on nonzero elements of F_2^n\{0} when n > 2, because it preserves planes. (if |S| = 4, we have G = S_4, K = S_3, H = V_4)

--- Other files

Corollary: A_n is simple for n \geq 5.

Proof: A_5 is simple.  All other A_n are (n-4)-transitive on n elements, with point stabilizer A_{n-1}.

Theorem: If G acts faithfully on n elements, and is n-transitive, then G is S_n.

Proof: If G is n-transitive, we get a surjection to S_n, and faithfulness means it is injective.

Theorem: If G acts faithfully on n elements and is (n-2)-transitive, then G is S_n or A_n.

Proof: Restricting to n-2 elements, we see that G contains S_{n-2}.  G is faithful, so it lies in S_n.  The order of G is a multiple of n(n-1)...3 = n!/2.
Intersection with A_n is index at most 2, and A_n has no index 2 subgroups ever.

2. Mathieu groups

Corollary: Mathieu groups M_n are simple, for n = 21,22,23,24.

Proof: M_{21} = PSL_3(F_4), which is simple.  The other M_n have M_{n-1} as point stabilizer and a doubly transitive action on n elements.  Index is not a prime power.

Aschbacher's construction: build a tower of Steiner systems S(21,5,2) < S(22,6,3) < S(23,7,4) < S(24,8,5).
A Steiner S(v,k,t) system is a pair (X,B), where X is a set of size v, and B is a collection of subsets of X, each of size k, such that for any size t subset Y of
X, there is a unique element of B containing Y.
Projective planes of order q are Steiner S(q^2+q+1,q+1,2) systems, where B is the set of lines.
From action of M_{24} on octads, get 11-dimensional representations over F_2: Golay code module and Tood code module.
Given a Steiner S(v,k,t) system (X,B) and a vertex x \in X, the residual Steiner system of (X,B) at x is (X-x, B(x)), where elements of B(x) are Y-x for x \in Y \in B.
If t \geq 2, then we get S(v-1,k-1,t-1).
Conversely, an extension of a Steiner S(v,k,t) system (X,B) is a Steiner S(v+1,k+1,t+1)-system (Z,A) such that (X,B) is the residual Steiner system at z, {z} = Z\X.
Given a Steiner S(v,k,t)-system, a subset I of X is independent if no (t+1)-subset of I is contained in a block of B.  E.g., for projctive planes, these are subsets with no collinear triples.
We write I_m(X) for the set of all independent subsets of size m.
An extension subset of X is a subset C of I_{k+1} such that each element of I_{t+1} is contained in a unique element of C.

Lemma: Let t > 1, X a Steiner S(v,k,t)-system, x \in X, and C(x) the set of blocks not containing x, and Y the residual Steiner system of X at x.  Then,
C(x) is an extension subset of Y.

Lemma: Let (X,B) be a Steiner S(v,k,t)-system, and C an extension subset of X.  Then:
  1. There is an extension Z of X such that C is the extension subset induced by C(z) for {z} = Z\X.
  2. Restriction to X identifies Aut(Z)_z with N(Aut X)(C).

Proof: The blocks of Z are elements of C and the unions of z with blocks of B.

We identify Aut(Z)_z with N_{Aut X}(C(z)).

***Extension hypothesis***

to appear

3. Golay 24-code. (Conway-Sloane p277)　　Maybe do general theory of linear codes for API?

shortest construction: Let \Omega = P^1(23).  Take Q = all squares mod 23, and consider all transformations under PGL_2(23).
These span an F_2-subspace C of P(\Omega), called the Golay code.

Generators of M_{24}:
\alpha: x \mapsto x+1.  (\infty) (0, 1, \ldots, 22)
\beta: x \mapsto 2x.  (\infty) (0) (1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12) (5, 10, 20, 17, 11, 22, 21, 19, 15, 7, 14)
\gamma: x \mapsto -1/x.  (0 \infty) (1 22) (2 11) (3 15) (4 17) (5 9) (6 19) (7 13) (8 20) (10 16) (12 21) (14 18)
\delta: x \mapsto x^3/9 on Q, and x \mapsto 9x^3 on \Omega\Q.  (\infty) (0) (1 18 4 2 6) (3) (5 21 20 10 7) (8 16 13 9 12) (11 19 22 14 17) (15)
PGL_2(23) is generated by \alpha, \beta, \gamma, and acts 3-transitively on \Omega, preserving the code.

Conway: 5-transitivity by direct calcluation.

Aschbacher starts with the Steiner system.

Define N = \Omega \ Q, N_\infty = Omega, and N_i = { n + i | n \in N}.  For any subset S \subset \Omega, define N_S = \sum_{i \in S} N_i

Theorem: C has dimension at most 12

Proof: N_\Omega = N_N = 0 (one checks that both have the form aN + bQ + c{\infty} + d{0} and matches coefficients), so all N_{N_i} = 0.
Then, N_C = 0 for all codewords.  Thus, if C is k-dimensional, we have at least k independentrelations between its generating sets N_i.
This implies k \leq 24-k.

Theorem: C has dimension at least 12.

Proof: N_{-2,0,2,3} = {0, 1, 2, 3, 4, 7, 10, 12} is a codeword with least element 0.
Translating this by 1-10 yields codewords with least element 1-10, spanning an 11-dimensional subspace.
Any other codeword containing \infty yields another dimension.

Theorem: M_{24} preserves C.

Theorem: Any nonempty codeword can be shrunk by taking symmetric difference with an octad.   Octads form a Steiner (5,8,24) system.
No codewords of weight less than 5.

Theorem: Stabilizer of an octad = 2^4.L_4(2)

Theorem: Order of M_{24}.

Theorem: Weight distribution.

4. N(A_1^{24}) lattice: The integral lattice (\sqrt{2}Z)^{24} has index 2^{24} in its dual lattice (2^{-1/2}Z)^{24}, with quotient group 2^{24}.
We get an even unimodular lattice by adjoining a 2^{12} subgroup of this quotient, such that all vectors have even norm and integral inner product with each other.
Evenness of the norm is equivalent to all codewords having weight 0 mod 4, and integral inner product follows from parallelogram rule (maybe?).
To get root system A_1^{24}, we need to have no norm 2 vectors outside the zero coset, i.e., no codewords of weight 4 or 20.
The Golay 24 code is the unique code satisfying these properties up to isomorphism.

5. Leech lattice

Consider the vector 1/4 C_\Omega.
Inner products with N(A_1^{24})-vectors are half-integers (1/8 times code weight), so the integral subset \Lambda_0 is an index 2 subgroup of N(A_1^{24}).
There is a unique unimodular lattice containing \Lambda_0 that is not N(A_1^{24}), and we call it Leech.
Leech has no roots: (proof?)
Leech has 196560 norm 4 vectors (proof: theta function is weight 12 index 1 with q-expansion 1 + 0q + nq^2, with n uniquely determined).
Weight 8 vectors form a system of orthogonal frames, which are taken disjointly to each other by isometries (proof?)
Stabilizer of a frame is a maximal subgroup of the form 2^{12}.M_{24}
Simplicity of Conway by using 13-Sylow and Frattini.

6. Central extensions - do we need Group cohomology?

extraspecial groups - classify.

double covers of lattices: commutator is (-1)^(a,b) - unique up to isomorphism.

Twisted group ring.

7. Vertex algebras.
Lattice construction - write it as twisted group ring tensor free boson space?  Or, consider intertwining operators between irred. Heisenberg modules?
How to feed :exp\sum -a_n/n z^n: into Lean?
Automorphism group of V_L: don't need full automorphism group, but do need diagonal octahedral action on V_{N(A_1^{24})}, and torus action on V_Leech

8. Construction of Monster vertex algebra.
Use triality: 4-group acts on V_{N(A_1^{24})} with S_3 conjugating.
Transport different copies of Leech pieces to V^\natural.
Multiplication is defined everywhere except on i-twisted \otimes j-twisted \to k-twisted.
For this, may need more symmetry to transport everything into Leech. (FLM does something like this)

9. Finiteness and Simplicity.
Follow ArXiv note.  Need elementary theory of algebraic groups to say that trivial Lie algebra implies finite.

10. Moonshine: need Virasoro representations and Borcherds-Kac-Moody Lie algebras and Adams operations.

11. Cummins-Gannon: More holomorphic functions.

12. More vertex algebra theory: C_1 tensor theory, C_2 modularity theory, Verlinde?