From now on we assume that G is the completion group of 𝒢 which is constrained at level 2. We identify the third geometric subgroup with the famous involution centralizer $$2_{+}^{1+12}\cdot 3\cdot \text{Aut}\left(M_{22}\right)$$ in J ₄. We also recover another important subgroup in G which is $$2^{11}:M_{24}.$$ This enables us to associate with G a coset geometry 𝒟(G) which eventually will be identified with the Ronan–Smith geometry for J ₄.

7.1 Encircling $2_{+}^{1+12}\cdot 3\cdot \text{Aut}\left(M_{22}\right)$

Let ϕ : 𝒢 → G be a faithful generating completion of the amalgam 𝒢 which is constrained at level 2. The existence of such a completion is guaranteed by (6.13.9). First we assume that ϕ : 𝒢 → G is universal among the completions which are constrained at level 2. Since the centre of N ^[2] = K ^[2] is trivial we can define such a completion in the following way (compare Section 2.5).

Let ϕ̃ : 𝒢 → G̃ be the universal completion of 𝒢, ϕ : 𝒢 → G be an arbitrary completion which is contrained at level 2, ψ : G̃ → G be the corresponding homomorphism of completions and Y be the kernel of ψ. Then the restriction of ψ to $${\tilde{G}}^{\left[2\right]}=〈\tilde{\varphi }\left(G^{\left[02\right]}\right),\tilde{\varphi }\left(G^{\left[12\right]}\right)〉$$ is a homomorphism onto $$G^{\left[2\right]}=〈\varphi \left(G^{\left[02\right]}\right),\varphi \left(G^{\left[12\right]}\right)〉$$ with kernel Y ^[2] = Y ∩ G̃^[2]. Since ϕ : 𝒢 → G is constrained at level 2, we have $$C_{G^{\left[2\right]}}\left(\varphi \left(N^{\left[2\right]}\right)\right)\le Z\left(\varphi \left(N^{\left[2\right]}\right)\right)=1.$$ On the other hand, the restriction of ψ to ϕ̃(N ^[2]) is an isomorphism onto ϕ(N ^[2]) and therefore, $$Y^{\left[2\right]}=C_{{\tilde{G}}^{\left[2\right]}}\left(\tilde{\varphi }\left(N^{\left[2\right]}\right)\right).$$ If we want G to be the ‘largest’ completion group subject to the property that it is constrained at level 2 we must take Y to be the smallest normal subgroup (p. 132 ) in G̃ which intersects G̃^[2] in Y ^[2]. This means that Y should be taken to be the normal closure in G̃ of C _G̃^[2] (ϕ̃(N ^[2])).

Alternatively we can define G to be the universal completion of the rank 3 amalgam $$𝒥=\left\{G^{\left[02\right]},G^{\left[1\right]},G^{\left[2\right]}\right\}.$$ It is worth mentioning that the existence of the amalgam 𝒥 is independent of the existence of completions of 𝒢 which are constrained at level 2. In fact, 𝒥 is the amalgam $$\left\{\tilde{\varphi }\left(G^{\left[0\right]}\right),\tilde{\varphi }\left(G^{\left[1\right]}\right),{\tilde{G}}^{\left[2\right]}\right\}$$ factorised over C _G̃^[2] (ϕ̃(N ^[2])). On the hand, 𝒥 possesses a faithful completion if and only if 𝒢 possesses a completion constrained at level 2.

From now on (unless explicitly stated otherwise) $$\varphi :𝒢\to G$$ is assumed to be an arbitrary faithful completion of 𝒢 which is constrained at level 2. The amalgam 𝒢 will be identified with its image in G under ϕ, so that we can plainly write $$G^{\left[i\right]}=〈G^{\left[0i\right]},G^{\left[1i\right]}〉.$$ By (4.2.6) and (4.2.8) N ^[2] and N ^[3] are non-trivial, so by (4.2.1 (iv)) G ^[2] and G ^[3] are proper subgroups in G. On the other hand by (5.4.1) N ^[4] = 1 and in Section 7.4 we will show that G ^[4] is in fact the whole of G.

Let Γ = Λ(𝒢, ϕ, G) be the coset graph corresponding to the completion ϕ: 𝒢 → G. Let x and {x, y} be defined as in the paragraph before (4.1.1) so that $$𝒢=\left\{G\left(x\right),G\left\{x,y\right\}\right\}.$$ Let Γ^[2] and Γ^[3] be the geometric subgraphs in Γ induced by the images of x under G ^[2] and G ^[3], respectively (compare (4.2.1 (iii))). Since Γ^[2] is of valency 3 and G ^[2] induces on the vertex set of Γ^[2] an action of G ^[2]/N ^[2] ≅ Sym₅ on the cosets of G ^[02]/N ^[2] ≅ Sym₃ × Sym₂ the following statement is an immediate consequence of the definition of the Petersen graph.

Lemma 7.1.1 Γ^[2] is isomorphic to the Petersen graph.

Since the action of G ^[3] on Γ^[3] is locally projective of type (3, 2) and Γ^[2] is a geometric cubic subgraph in Γ^[3], (7.1.1), (5.2.3), and (11.4.3) imply the following.

Lemma 7.1.2 One of the following two possibilities takes place:

1. (i) Γ^[3] is the octet graph Γ(M ₂₂), C _{G ^[3]} (N ^[3]) = Z(N ^[3]) = Z ^[3] ≅ 2, G ^[3]/N ^[3] ≅ Aut (M ₂₂) and $$G^{\left[3\right]}\cong 2_{+}^{1+12}\cdot 3\cdot \text{Aut}\left(M_{22}\right);$$

(p. 133 )

2. (ii) Γ^[3] is the Ivanov–Ivanov–Faradjev graph Γ(3 · M ₂₂), C _{G ^[3]} (N ^[3]) ≅ 2 × 3, G ^[3]/N ^[3] ≅ 3 · Aut (M ₂₂) and $$G^{\left[3\right]}\cong \left(2_{+}^{1+12}\times 3\right)\cdot 3\cdot \text{Aut}\left(M_{22}\right).$$

It will be proved in Section 7.4 that the possibility (7.1.2 (i)) takes place. Clearly G ^[3] is a completion of the amalgam $${𝒢}^{\left[3\right]}=\left\{G^{\left[03\right]},G^{\left[13\right]},G^{\left[23\right]}\right\}.$$ Since the completion ϕ : 𝒢 → G is constrained at level 2, it is rather straightforward to check that $$C_{G^{\left[2\right]}}\left(N^{\left[3\right]}\right)\le Z\left(N^{\left[3\right]}\right)=Z^{\left[3\right]}$$ and therefore the amalgam 𝒢̄^[3] = {G ^[03]/N ^[3], G ^[13]/N ^[3], G ^[23]/N ^[3]} is isomorphic to the amalgam $\widehat{{𝒜}^{\left[3\right]}}$ defined before (5.2.1). Hence by (5.2.1) 𝒢̄^[3] is isomorphic to the amalgam 𝒵 as in (11.4.1).

Lemma 7.1.3 Let C ^[3] be the universal completion of 𝒢^[3]. Then C ^[3]/N ^[3] is the universal completion of 𝒢̄^[3] ≅ 𝒵, therefore C ^[3]/N ^[3] ≅ 3 · Aut (M ₂₂).

Proof Let K̃ ≅ 3 · Aut (M ₂₂) be the universal completion of 𝒵 and let us identify 𝒵 with its image in K̃. Let α be a homomorphism of 𝒢^[3] onto 𝒵 which is the composition of the canonical homomorphism g ↦ gN ^[3] of 𝒢^[3] onto 𝒢̄^[3] and an isomorphism of 𝒢̄^[3] onto 𝒵. Let $$C^{\left[3\right]}\times \tilde{K}=\left\{c,k|c\in C^{\left[3\right]},k\in \tilde{K}\right\}$$ be the direct product of C ^[3] and K̃ and let χ be the subset of C ^[3] × K̃ consisting of the pairs (c, k), such that α(c) = k. Then χ is isomorphic to 𝒢^[3]. Furthermore if X is the subgroup in C ^[3] × K̃ generated by χ then the restriction to X of the canonical homomorphism of C ^[3] × K̃ onto K̃ is surjective and the claim follows. ■

By (7.1.3) if G ^[3] is the universal completion of 𝒢^[3] then the possibility (ii) in (7.1.2) takes place. Therefore there is no way we can get down to the possibility (i) looking at the amalgam 𝒢 only and some further subgroups of G should be brought into play.

7.2 Tracking 2¹¹ : M ₂₄

In Sections 3.7 and 3.8 we have seen that G ^[1] is a semidirect product of Q ^[m] ≅ 2¹¹ and A ^[1] L ^[1] ≅ 2⁴ : L ₄(2). The relevant action is isomorphic to the action of the octad stabilizer in M ₂₄ on the irreducible Todd module 𝒞₁₁. In Section 7.4 we will prove the following.

Proposition 7.2.1 Let G ^[m] be the subgroup in G generated by the normalisers of Q ^[m] in G ^[1], G ^[2], and G ^[3]. Then $$G^{\left[m\right]}\cong 2^{11}:M_{24},$$ (p. 134 ) more specifically G ^[m] is the semidirect product of Q ^[m] ≅ 𝒞̄₁₁ and M ₂₄ with respect to the natural action.

For i = 1, 2 and 3 put G ^[mi] = N _{G ^[i]} (Q ^[m]), 𝒢^[m] = {G ^[mi] | 1 ≤ i ≤ 3} and 𝒢̄^[m] = {G ^[mi]/Q ^[m] | 1 ≤ i ≤ 3}, so that 𝒢̄^[m] is the quotient of 𝒢^[m] over Q ^[m]. Notice that G ^[m1] = G ^[m].

Lemma 7.2.2 The following assertions hold:

1. (i) G ^[m1]/Q ^[m] ≅ 2⁴ : L ₄(2)

2. (ii) G ^[m2]/Q ^[m] ≅ 2⁶ : (L ₃(2) × Sym₃);

3. (iii) either

   1. (a) (7.1.2 (i) takes place and G ^[m3]/Q ^[m] ≅ 2⁶:3 · Sym₆, or

   2. (b) (7.1.2 (ii)) takes place and G ^[m3]/Q ^[m] ≅ (2⁶ × 3):3 · Sym₆.

Proof Statement (i) follows is directly from (3.8.1). In order to establish (ii) we locate Q ^[m] inside G ^[2]. It is clear that Q ^[m] is contained in G ^[2] (for instance because [G ^[1] : G ^[12] = 15 is odd and Q ^[m] is a normal 2-subgroup in G ^[1]). By (4.2.7) |N ^[2] ∩ Q ^[m]| = 2⁹ and the image of Q ^[m] in G ^[2]/N ^[2] ≅ Sym₅ is an elementary abelian subgroup of order 4, which stabilizes an edge of Γ^[2] as a whole but not vertexwisely. This means that Q ^[m] N ^[2]/N ^[2] is contained in the commutator subgroup of G ^[2]/N ^[2], isomorphic to Alt₅.

Let S ₇ be a Sylow 7-subgroup in G ^[2], C ≅ Sym₅ be the complement to S ₇ in C_G ^[2](S ₇) (compare (4.9.1)) and R be the elementary abelian subgroup of order 4 in C, such that RN ^[2]/N ^[2] = Q ^[m] N ^[2]/N ^[2]. We claim that R is contained in Q ^[m]. In fact, by (3.7.1) G ^[1]/Q ^[m] ≅ 2⁴ : L ₄(2) and since [G ^[1] : G ^[12]] = 15 is not divisible by 7, Q ^[m] is normalized by a Sylow 7-subgroup in G ^[2]. By Sylow's theorem without loss we assume that this subgroup is S ₇. By (4.9.1) (ii) C _{Q ^[2]} (S ₇) = 1. Therefore $$C_{Q^{\left[m\right]}}\left(S_7\right)N^{\left[2\right]}/N^{\left[2\right]}=Q^{\left[m\right]}{N}^{\left[2\right]}/N^{\left[2\right]}$$ and the claim follows. Next we claim that Q ^[m] = RC _{Q ^[2]} (R). Since R ≤ Q ^[m] and Q ^[m] is abelian, Q ^[m] is obviously in the centralizer of R in Q ^[2] and hence we only have to show that |C _{Q ^[2]} (R)| is at most 2⁹. The subgroup C _{G ^[2]} (R) contains S ₇, therefore C _{Q ^[2]} (R) is normalized by S ₇. Clearly C _{Q ^[2]} (R) contains Z ^[2] and therefore every dent of Q ^[2] is either completely contained in C _{Q ^[2]} (R) or intersects C _{Q ^[2]} (R) in Z ^[2]. In addition, since R commutes with S ₇ and every dent is the direct sum of two non-isomorphic S ₇-modules, whenever R normalizes a dent, it necessarily centralizes it. Now it only remains to recall that by (4.7.4) C acts on the set of dents as it acts on the edge-set of the Petersen graph Γ^[2]. Finally, R stabilizes exactly three edges of Γ^[2] (these edges form the antipodal triple containing {x, y}).

By the above paragraph the number of conjugates of Q ^[m] in G ^[2] is equal to the number of conjugates of R in C (which is five). Since $$N_C\left(R\right)\cong {\text{Sym}}_4\cong 2^2:{\text{Sym}}_3,$$ (ii) follows.

(p. 135 ) The stabilizer in G of an edge e = {u, v} of Γ is a conjugate of G ^[1] and by (3.7.1) this stabilizer contains a unique normal elementary abelian subgroup Q_e of order 2¹¹, which is of course a conjugate of Q ^[m]. By the above paragraph whenever two edges e and f are contained in a common geometric cubic subgraph Σ ≅ Γ^[2] and are antipodal in the line graph of Σ, the equality $$Q_e=Q_f$$ holds (notice that there are 15 edges in Σ and only 5 different conjugates of Q ^[m] in G ^[2]). Let Φ = Φ_Γ be the local antipodality graph of Γ, so that Φ is a graph on the edge-set of Γ in which two edges are adjacent if they are contained in a common geometric cubic Petersen subgraph Σ and are antipodal in the line graph of Σ. Then Q_e = Q_f whenever e and f are in the same connected component of Φ.

Let us turn to (iii). It is clear that Q ^[m] ≤ G ^[3]. On the other hand, since $Q^{\left[3\right]}=O_2\left(G^{\left[3\right]}\right)\cong 2_{+}^{1+12}$ is extraspecial, while Q ^[m] is elementary abelian, |Q ^[m] ∩ Q ^[3]| ≤ 2⁷ by (1.6.7). Let Ψ = Φ_Γ^[3] be local antipodality graph of Γ^[3] and Ψ ^c be the connected component of Ψ containing {x, y}. Since Γ^[3] is either the octet graph or the Ivanov–Ivanov–Faradjev graph, by (11.4.4) and the paragraph after that lemma Ψ ^c contains 15 or 45 edges of Γ^[3] depending on whether we are in case (a) or (b). By the above paragraph the stabilizer S of Ψ ^c in G ^[3] is contained in G ^[m3]. Furthermore, S contains N ^[3] and S/N ^[3] is 2⁴ : Sym₆ and (2⁴ × 3) · Sym₆ in the respective cases (a) and (b). Since Q ^[m] N ^[3]/N ^[3] = O ₂(S/N ^[3]), using the well-known fact that K_h = N_K (O ₂(K_h )) for the stabilizer K_h ≅ 2⁴ : Sym₆ of a hexad in K ≅ Aut (M ₂₂), we conclude that S is the whole of G ^[m3], which completes the proof of (iii). ■

Lemma 7.2.3 Suppose that (7.1.2. (i)) takes place. Then

1. (i) the coset geometry corresponding to the embedding into G ^[m]/Q ^[m] of the amalgam $$\left\{G^{\left[mi\right]}/Q^{\left[m\right]}|1\le i\le 3\right\}$$ is described by the locally truncated diagram



2. (ii) G ^[m]/Q ^[m] ≅ M ₂₄;

3. (iii) G ^[m] splits over Q ^[m].

Proof First notice that the assertion (7.2.2 (iii) (a)) holds. Calculating the intersections of the G ^[mi]'s we obtain (i). Now (ii) is by (i) and (11.2.1), while (iii) is by (ii), (3.7.1) and Gaschütz's theorem. ■

(p. 136 ) 7.3 P-geometry of G ^[4]

In this section for a subsequence α of 0123 we denote the subgroup G ^[α4] by F ^[α]. This convention also applies when α is empty, so that F = G ^[4]. Let ℱ = {F ^[0], F ^[1]} be the corresponding subamalgam in F, let Ξ = Λ^[4] = Λ(𝒢^[4], ϕ^[4], F) be the coset graph associated with the completion $${\varphi }^{\left[4\right]}:{𝒢}^{\left[4\right]}\to F$$ (which is the restriction of ϕ to 𝒢^[4]). At this stage we do not know yet that F is the whole of G, but at any event the action of F on Ξ is faithful (since N ^[4] is trivial by (5.4.1)) and locally projective of type (4, 2). Let {u, v} be the edge of Ξ such that $$ℱ=\left\{F\left(u\right),F\left\{u,v\right\}\right\},$$ where ℱ is identified with its image in F under ϕ^[4]. For i = 2 and 3 let Ξ^[i] be the geometric subgraph in Ξ induced by the images of u under F ^[i] and let I ^[i] be the vertexwise stabilizer of Ξ^[i] in F

Lemma 7.3.1 The following assertions hold:

1. (i) F ^[0] = G ^[04] ≅ 2⁴⁺⁴ : 2⁶ : L ₄(2);

2. (ii) $F^{\left[1\right]}=〈G^{\left[014\right]},t_1〉\cong 2^{6+5+6}.\left(L_3\left(2\right)\times 2\right)\cong 2^{11}:2_{+}^{1+6}:L_3\left(2\right);$;

3. (iii) F ^[2] ≅ 2³⁺¹² · (sym₄ × Sym₅), Ξ^[2] is the Petersen and I ^[2] ≅ 2²⁺¹² × Sym₄;

4. (iv) $F^{\left[3\right]}\cong 2_{+}^{1+12}\cdot 3\cdot \text{Aut}\left(M_{22}\right)$, Ξ^[3] is the Ivanov–Ivanov–Faradjev graph and $I^{\left[3\right]}\cong 2_{+}^{1+12}$.

Proof Since F ^[0] and F ^[1] are the stabilizers of U ₁ in G ^[0] and G ^[1], respectively (i) and (ii) are quite clear. In terms of Section 3.8 F ^[1] is a semidirect product of Q ^[m] ≅ 2¹¹ and the stabilizer of U ₁ in A ^[1] L ^[1] ≅ 2⁴ : L ₄(2). The latter stabilizer coincides with the centralizer of a central involution in L ^[0] ≅ L ₅(2), isomorphic to $2_{+}^{1+6}:L_3\left(2\right)$. We know that F ^[2] is the subgroup in G ^[2] generated by G ^[024] and G ^[124]. The set 𝒫 of geometric subgraphs of valency 7 in Γ containing Γ^[2] is of size 7 (of course Γ^[3] ∈ 𝒫). The action (isomorphic to L ₃(2)) of N ^[2] on 𝒫 induces a structure of the projective plane of order 2. Then F ^[2] is the stabilizer in G ^[2] of a line in that projective plane structure which gives (iii).

The subgroup F ^[3] is generated by G ^[034] and G ^[134]. Since Q ^[3] = O ₂(N ^[3]) = N ^[3] ∩ G ^[014] we immediately conclude that $$I^{\left[3\right]}\ge Q^{\left[3\right]}\cong 2_{+}^{1+12}.$$ On the other hand, the whole of N ^[3] could not be in I ^[3] since it is not even in G ^[014]. The action of F ^[3] on Ξ^[3] is locally projective of type (3, 2) and by (ii) Ξ^[2] is a geometric cubic subgraph in Ξ^[3] isomorphic to the Petersen graph. By (11.4.3) this implies that Ξ^[3] is either the octet graph Γ(M ₂₂) of the (p. 137 ) Ivanov–Ivanov–Faradjev graph Γ(3 · M ₂₂). Let $$\chi :G^{\left[3\right]}\to \text{Out}{Q}^{\left[3\right]}$$ be the natural homomorphism. By (5.2.4) the image of χ is isomorphic to 3 · Aut (M ₂₂). For α = 0 and 1 the subgroups G ^[α34] and $N^{\left[3\right]}\cong 2_{+}^{1+12}:3$ factorize G ^[α3] and hence by (5.2.1 (i), (ii)) we have $$\chi \left(F^{\left[03\right]}\right)\cong 2^3:L_3\left(2\right)\times 2,\chi \left(F^{\left[13\right]}\right)\cong \left(2_{+}^{1+4}:{\text{Sym}}_3\times 2\right)\cdot 2.$$ Since χ(G ^[3]) does not split over O ₃(χ(G ^[3])), $$3\cdot \text{Aut}\left(M_{22}\right)\cong \chi \left(G^{\left[3\right]}\right)=\chi \left(F^{\left[3\right]}\right)=〈\chi \left(F^{0\left[3\right]}\right),\chi \left(F^{\left[13\right]}\right)〉$$ and therefore F ^[3]/I ^[3] possesses a homomorphism onto 3 · Aut (M ₂₂). Thus (iv) follows. ■

Let 𝒢(G ^[4]) be the geometry, whose elements of type 1 are the vertices of Ξ, the elements of type 2 are the edges of Ξ, the elements of type 3 are the geometric cubic subgraphs in Ξ and the elements of type 4 are the geometric subgraphs of valency 7 in Ξ; the incidence relation is via inclusion. As a direct consequence of (7.3.1) we obtain the following

Proposition 7.3.2 The geometry 𝒢(G ^[4]) is a P-geometry of rank 4 with the diagram

In Section 7.4 we will show that G ^[4] is the whole of G and by the Main Theorem the latter is J ₄. Therefore the geometry in (7.3.2) is the P-geometry 𝒢(J ₄) of J ₄ first constructed in (Ivanov 1987).

For 1 ≤ i ≤ 3 put F ^[mi] = F ^[i] ∩ Q ^[m] and F ^[m] = 〈F ^[mi] | 1 ≤ i ≤ 3〉.

Lemma 7.3.3 The following assertions hold:

1. (i) $F^{\left[m1\right]}/Q^{\left[m\right]}\cong 2_{+}^{1+6}:L_3\left(2\right)$ and F ^[m1] splits over Q ^[m];

2. (ii) F ^[m2]/Q ^[m] ≅ 2⁶ : (Sym₄ × Sym₃);

3. (iii) F ^[m3]/Q ^[m] ≅ 2⁶ : 3 · Sym₆;

4. (iv) the coset geometry ℳ corresponding to the embedding of the amalgam {F ^[mi]/Q ^[m] | 1 ≤ i ≤ 3} into F ^[m]/Q ^[m] is described by the tilde diagram



5. (v) F ^[m]/Q ^[m] ≅ M ₂₄ and ℳ ≅ 𝒢(M ₂₄;

6. (vi) Q ^[m] is the irreducible Todd module 𝒞̄₁₁;

7. (vii) F ^[m] splits over Q ^[m].

(p. 138 ) Proof A mere comparison of (7.2.2) and (7.3.1) gives (i) to (iii). The diagram of ℳ can be recovered by direct calculating the intersection of the F ^[mi]'s. Alternatively one can employ the following combinatorial realization of ℳ. Let ϒ = Φ_Ξ be the local antipodality graph of Ξ. Then, arguing as in the proof of (7.2.2), one can see that F ^[m] coincides with the stabilizer in F of the connected component ϒ ^c of ϒ containing {u, v}. The elements of ℳ are the vertices of ϒ ^c (which are edges of Ξ), the intersections of the vertex set of ϒ ^c with edge-sets of geometric subgraphs of valency 3 and 7. Then by (7.3.2) and the paragraph after (11.4.4) we obtain the desired diagram.

By (11.2.2) the assertions (i) to (iv) imply that F ^[m]/Q ^[m] is either M ₂₄ or H e. Since Q ^[m] is a non-trivial module in which F ^[m3] stabilizes the 1-dimensional subspace Z ^[3] the latter possibility is excluded, since the index of 2⁶ : 3 · Sym₆ in H e is 29, 155 (cf. Conway et al. (1985) and Section 11.2 and hence (v) follows. The subgroup F ^[m2] stabilizes in Q ^[m] the 2-dimensional subspace Z ^[2] which contains the 1-dimensional subspace Z ^[3] stabilized by F ^[m3]. In terms of Ivanov and Shpectorov (2002) this means that Q ^[m] is a quotient of the universal representation group of ℳ ≅ 𝒢(M ₂₄), so that (vi) follows from Proposition 4.3.1 in Ivanov and Shpectorov (2002). Finally (vii) follows from (i) in view of Gaschütz's theorem. ■

It is worth mentioning that the proof of (7.3.3 (v)) is the only place in the present volume where we essentially make use of a result (which is (11.2.2)) whose proof relies on computer-aided calculations.

7.4 G ^[4] = G

First we show that G ^[m4] = G ^[m] (recall that G ^[m4] ≅ 2¹¹ : M ₂₄ by (7.3.3 (v), (vi), (vii)).

Lemma 7.4.1 Suppose that G ^[m4] ≅ 2¹¹ : M ₂₄ is a proper subgroup in G ^[m] Then the coset geometry 𝒩 corresponding to the embedding into G ^[m]/Q ^[m] of the amalgam $$\left\{G^{\left[mi\right]}/Q^{\left[m\right]}|1\le i\le 4\right\}$$ is described by the rank 4 tilde diagram

Proof We claim that under the hypothesis (7.1.2 (ii)) takes place. In fact, otherwise $$G^{\left[m\right]}\cong G^{\left[m4\right]}\cong 2^{11}:M_{24}$$ by (7.2.3), (7.3.3 (v), (vi), (vii)) and the order comparison. Then the structure of the G ^[mi]'s can be read from (7.2 (i), (ii), (iii) (b)), and (7.3 (v), (vi), (vii)). Calculating the intersections we get the diagram. ■

(p. 139 ) Lemma 7.4.2 G ^[m4] = G ^[m].

Proof If the claim fails then by (7.4.1) G ^[m]/Q ^[m] acts flag-transitively on a rank 4 tilde geometry 𝒩. In terms of Ivanov and Shpectorov (2002) this geometry is of truncated M ₂₄-type and it does not exist by Proposition 12.4.6 and 12.5.1 in Ivanov and Shpectorov (2002). ■

Proof of Proposition 7.2.1 The result is now immediate by (7.3.3) and (7.4.2). ■

Lemma 7.4.3 The possibilities (7.1.2 (i)) and (7.2.2 (iii) (a)) take place, so that $$G^{\left[3\right]}\cong 2_{+}^{1+12}\cdot 3\cdot \text{Aut}\left(M_{22}\right)\text{and}{G}^{\left[3m\right]}/Q^{\left[m\right]}\cong 2^6:3\cdot {\text{Sym}}_6.$$

Proof By (7.2.1) G ^[m]/Q ^[m] ≅ M ₂₄ and the latter group just does not contain subgroups as in (7.2.2 (iii) (b)) already by Lagrange theorem. ■

We are ready to prove the main result of the section.

Proposition 7.4.4 G ^[4] = G.

Proof By (7.4.1) G ^[m] = G ^[m4] ≤ G ^[4]. Also G ^[1] ≥ G ^[4] since G ^[1] ≥ G ^[m] (as remarked before (7.2.2)). But G ^[0] is generated by G ^[01] and G ^[04], and so also G ^[0] ≥ G ^[4]. This clearly implies G ^[4] = G. ■

We refer the reader to sections 9.5, 9.6 in Ivanov (1999) for general discussion about the existence/non-existence of geometric subgraphs.

Lemma 7.4.5 Let Y be a Sylow 3-subgroup in O _2,3(G ^[3]). Then

1. (i) C _{G ^[3]} (Y) ≅ 6 · M ₂₂ is a non-split central extension of a cyclic group of order 6 by M ₂₂;

2. (ii) G ^[3] does not split over $Q^{\left[3\right]}=O_2\left(G^{\left[3\right]}\right)\cong 2_{+}^{1+12}$.

Proof Since Y is a Sylow 3-subgroup of N ^[2] the result is by (4.9.1 (iii)). ■

By (7.4.5) the Schur multiplier of M ₂₂ possesses the cyclic group of order 6 as a factor-group. In 1976, when (Janko 1976) was published this cyclic group was believed to be the whole Schur multiplier of M ₂₂. In (Mazet 1979) the multiplier of M ₂₂ was proved to be the cyclic group of order 12.

7.5 Maximal parabolic geometry 𝒟

We start this section by summarizing the information about the action of G on Γ we have obtained so far.

(p. 140 ) Proposition 7.5.1 Let G be a completion of the amalgam 𝒢 which is constrained at level 2 and let Γ be the coset graph associated with this completion. Then

1. (i) Γ is connected of valency 31 and the action of G on Γ is locally projective of type (5, 2);

2. (ii) G(x) = G ^[0] ≅ 2¹⁰ : L ₅(2);

3. (iii) G{x, y} = G ^[1] ≅ 2⁶⁺⁴⁺⁴ · (L ₄(2) × 2) ≅ 2¹¹ : 2⁴ : L ₄(2);

4. (iv) the geometric cubic subgraph Γ ^[2] is isomorphic to the Petersen graph and $$G\left\{{\Gamma }^{\left[2\right]}\right\}=G^{\left[2\right]}\cong 2^{3+12}\cdot \left(L_3\left(2\right)\times {\text{Sym}}_5\right);$$

5. (v) the geometric subgraph Γ^[3] of valency 7 is isomorphic to octet graph and $$G\left\{{\Gamma }^{\left[3\right]}\right\}=G^{\left[3\right]}\cong 2_{+}^{1+12}\cdot 3\cdot \text{Aut}\left(M_{22}\right);$$

6. (v) there are no geometric subgraphs of valency 15 and G ^[4] ≔ 〈G ^[04], G ^[14]〉 is the whole of G;

7. (vi) if Φ = Φ_Γ is the local antipodality graph of Γ and Φ ^c is the connected component of Φ containing {x, y} then Φ ^c is isomorphic to the octad graph Γ(M ₂₄) and $$G\left\{{\Phi }^c\right\}=G^{\left[m\right]}\cong 2^{11}:M_{24}.$$

Proof (i) and (ii) are already in (4.1.1), (iii) is by (7.1.1), (iv) is by (7.4.3), (v) is by (7.4.4). Finally (vi) is by (7.2.3) since (7.1.2 (i)) takes place by (7.4.3). ■

Let ℱ(G) be a geometry such that

1. (0) the elements of type 0 are the vertices of Γ;

2. (1) the elements of type 1 are the edges of Γ;

3. (2) the elements of type 2 are the geometric cubic subgraphs;

4. (3) the elements of type 3 are the geometric cubic subgraphs of valency 7;

5. (i) the incidence relation is via inclusion.

Then it is immediate from (7.5.1) that ℱ(G) belongs to the locally truncated Petersen diagram

By the Main Theorem G ≅ J ₄ so ℱ(G) is another geometry for J ₄ constructed in (Ivanov 1987).

More fruitful for our current purposes is the geometry 𝒟 = 𝒟(G) whose elements are as in ℱ(G), only instead of the elements of type 1 (which are the edges of Γ) we take elements of type m which are the connected components of the local antipodality graph Φ of Γ. The incidence relation between the elements of type 0, 2, and 3 is as in ℱ(G). A connected component of Φ (an element of type m) is adjacent to an element f ∈ ℱ(G) if f is incident in ℱ(G) to an edge of Γ contained in that connected component.

(p. 141 ) Since G is generated by G ^[0] and G ^[1] it is a standard result that both ℱ(G) and 𝒟(G) are connected.

7.6 Residues in 𝒟

Let 𝒟 = 𝒟(G) be the geometry defined in Section 7.5. Recall that the set of types of 𝒟 is {m, 0, 2, 3}. For i ∈ {m, 0, 2, 3} the set of elements of type i on 𝒟 will be denoted by 𝒟^[i]. Often we will write the type of an element above its name, for instance we write $\stackrel{3}{d}$ for an element d of type 3. The stabilizer in G of this element will be denoted by $G_d^{\left[3\right]}$. The residue in 𝒟 of an element a (whose type will be clear from the context) will be denoted by 𝒟 _a .

Recall that a path in 𝒟 is a sequence π = (a ₀, a ₁, a ₂, a ₃,…,a_s ) of its elements such that a_i is incident to a _i+1 but neither equal nor incident to a _i+2 for every 1 ≤ i ≤ s −1. In this case s is the length of π.

With every element $\stackrel{i}{a}\in 𝒟$ we associate a certain combinatorial/geometrical structure (whose isomorphism type depends on i only). Then the residue 𝒟 _a of a in 𝒟 and the stabilizer $G_a^{\left[i\right]}$ of a in G possess natural descriptions in terms of this structure. This works in the following way:

Type m: If $\stackrel{m}{a}$ is an element of type m then there is a Witt design $W_a^{\left[24\right]}$ of type S(5, 8, 24). If ℬ _a , 𝒯 _a and 𝒮 _a are the octads, trios and sextets of $W_a^{\left[24\right]}$, then $${𝒟}_a^{\left[0\right]}={ℬ}_a\times GF\left(2\right),{𝒟}_a^{\left[2\right]}={𝒯}_a,{𝒟}_a^{\left[3\right]}={𝒮}_a.$$ The incidence relation in 𝒟 _a is via the refinement relation on the corresponding partitions of the element set of $W_a^{\left[24\right]}$. For instance suppose that $\left(B,\alpha \right)\in {𝒟}_a^{\left[0\right]}$ and $𝒮\in {𝒟}_a^{\left[3\right]}$, where B is an octad of $W_a^{\left[24\right]}$ (identified with the partition of the set of 24 elements into the octad B and its complement), α ∈ GF(2) and S is a sextet. Then (B, α) and S are incident if and only if B is the union of two tetrads from S. In particular, α does not effect the incidence. The stabilizer $G_a^{\left[m\right]}$ is the semidirect product of the automorphism group $M_a^{\left[m\right]}\cong M_{24}$ of $W_a^{\left[24\right]}$ and the irreducible Todd module $Q_a^{\left[m\right]}\cong {\overline{𝒞}}_{11}$. The module $Q_a^{\left[m\right]}$ is considered as a section of the GF(2)-permutation module of $M_a^{\left[m\right]}$ on the set of elements of $W_a^{\left[24\right]}$. In particular $M_a^{\left[m\right]}$ has two orbits on the set of non-zero vectors in $Q_a^{\left[m\right]}$; the elements in one of the orbits are indexed by pairs of elements of $W_a^{\left[24\right]}$, while those from the other orbit are indexed by the sextets from 𝒮 _a . Dually, the hyperplanes in $Q_a^{\left[m\right]}$ are indexed by the octads from ℬ _a and by the complementary pairs of dodecads. The subgroup $Q_a^{\left[m\right]}\cong O_2\left(G_a^{\left[m\right]}\right)$ is the kernel of the action of $G_a^{\left[m\right]}$ on ${𝒟}_a^{\left[2\right]}\cup {𝒟}_a^{\left[3\right]}$. Every orbit $Q_a^{\left[m\right]}$ on ${𝒟}_a^{\left[0\right]}$ is of the form {(B, 0), (B, 1)}, where B ∈ ℬ _a . An element $q\in Q_a^{\left[m\right]}$ fixes this orbit elementwise if and only if q is in the hyperplane corresponding to B. For every α ∈ GF(2) the complement $M_a^{\left[m\right]}$ stabilizes {(B, α) | B ∈ ℬ _a } as a whole and acts on it as it acts on ℬ _a .

(p. 142 ) Type 0: If $\stackrel{0}{b}$ is an element of type 0 then there is a 5-dimensional vector space V_b over GF(2) such that $${𝒟}_b^{\left[3\right]}=\left[\begin{array}{l}{V}_b\\ 2\end{array}\right],{𝒟}_b^{\left[2\right]}=\left[\begin{array}{l}{V}_b\\ 3\end{array}\right],{𝒟}_b^{\left[m\right]}=\left[\begin{array}{l}{V}_b\\ 4\end{array}\right],$$ (where $\left[\begin{array}{l}{V}_b\\ i\end{array}\right]$ stands for the set of i-dimensional subspaces in V_b ). The incidence in 𝒟 _b is by inclusion. The subspace corresponding to an element x in 𝒟 _b will be denoted by V_b (x). The stabilizer $G_b^{\left[0\right]}$ is the semidirect product with respect to the natural action of the general linear group $L_b^{\left[0\right]}\cong L_5\left(2\right)$ and the exterior square $Q_b^{\left[0\right]}\cong 2^{10}$. The latter is the kernel of the action of $G_b^{\left[0\right]}$ on 𝒟 _b , while $G_b^{\left[0\right]}/Q_b^{\left[0\right]}\cong L_b^{\left[0\right]}$ acts in the natural way. If b is the vertex x of Γ as in (7.5.1 (ii)) then V_b = U ₅, $G_b^{\left[0\right]}=G^{\left[0\right]}$ etc.

Type 2: If $\stackrel{2}{c}$ is an element of type 2 then there is a Petersen graph Θ _c and a 3-dimensional GF(2)-vector space $Z_c^{\left[2\right]}$ such that $${𝒟}_c^{\left[3\right]}=\left[\begin{array}{l}{Z}_c^{\left[2\right]}\\ 1\end{array}\right],{𝒟}_c^{\left[0\right]}=V\left({\Theta }_c\right),$$ and ${𝒟}_c^{\left[m\right]}$ is the set of antipodal triples of edges of Θ _c (considered also as 6-element subsets of V(Θ _c )). Every element from ${𝒟}_c^{\left[3\right]}$ is incident to every element from ${𝒟}_c^{\left[0\right]}\cup {𝒟}_c^{\left[m\right]}$ while the incidence between the elements from ${𝒟}_c^{\left[0\right]}$ and the elements from ${𝒟}_c^{\left[m\right]}$ is via inclusion. The stabilizer $$G_c^{\left[2\right]}\cong 2^{3+12}\cdot \left(L_3\left(2\right)\times {\text{Sym}}_5\right)$$ is isomorphic to the pentad group. Furthermore, $Q_c^{\left[2\right]}=O_2\left(G_c^{\left[2\right]}\right)$ is the kernel of the action of $G_c^{\left[2\right]}$ on 𝒟 _c $N_c^{\left[2\right]}\cong 2^{3+12}:L_3\left(2\right)$ is the kernel of the action of $G_c^{\left[2\right]}$ on Θ _c . $L_c^{\left[2\right]}\cong L_3\left(2\right)$ be a complement to $Q_c^{\left[2\right]}$ in $N_c^{\left[2\right]}$ and let $S_c^{\left[2\right]}\cong {\text{Sym}}_5$ be a complement to $N_c^{\left[2\right]}$ in $G_c^{\left[2\right]}$ (recall that $Q_c^{\left[2\right]}$ is not complemented in $G_c^{\left[2\right]}$. If c is the geometric cubic subgraph Γ^[2] in Γ as in (7.5.1 (iv)), then $G_c^{\left[2\right]}=G^{\left[2\right]}$, $Q_c^{\left[2\right]}=Q^{\left[2\right]}$ etc.

Type 3: If $\stackrel{3}{d}$ is an element of type 3 then there is a Witt design $W_d^{\left[22\right]}$ of type S(3, 6, 22) associated with d. If 𝒪 _d is the set of octets, ℋ _d is the set of hexads and 𝒫 _d is the set of pairs in $W_d^{\left[22\right]}$ then $${𝒟}_d^{\left[0\right]}={𝒪}_d,{𝒟}_d^{\left[m\right]}={\mathscr{H}}_d,{𝒟}_d^{\left[2\right]}={𝒫}_d$$ with the incidence relation as in the geometry ℋ(M ₂₂). The stabilizer $G_d^{\left[3\right]}$ is of the form $$G_d^{\left[3\right]}\cong 2_{+}^{1+12}\cdot 3\cdot \text{Aut}\left(M_{22}\right)$$ (p. 143 ) and $N_d^{\left[3\right]}=O_{2,3}\left(G_d^{\left[3\right]}\right)$ is the kernel of the action of $G_d^{\left[3\right]}$ on the residue 𝒟 _d . Let $M_d^{\left[3\right]}\cong 6\cdot \text{Aut}\left(M_{22}\right)$ be the normalizer in $G_d^{\left[3\right]}$ of a Sylow 3-subgroup $Y_d^{\left[3\right]}$ in $N_d^{\left[3\right]}$ (compare (7.4.5)). Then $M_d^{\left[3\right]}\cap Q_d^{\left[3\right]}=Z_d^{\left[3\right]}$ and $Q_d^{\left[3\right]}{M}_d^{\left[3\right]}=G_d^{\left[3\right]}$ (where $Q_d^{\left[3\right]}=O_2\left(G_d^{\left[3\right]}\right)$ and $Z_d^{\left[3\right]}=Z\left(Q_d^{\left[3\right]}\right)$). If d is the geometric subgraph Γ^[3] of valency 7 in Γ as in (7.5.1 (v)) then $G_d^{\left[3\right]}=G^{\left[3\right]},N_d^{\left[3\right]}=N^{\left[3\right]}$ etc.

It is immediate from the above that 𝒟(G) belongs to the following diagram (cf. Section 10.4 for the definitions of the relevant rank 2 residues). Instead of types next to every node we indicate the structure of the corresponding stabilizer in G.

7.7 Intersections of maximal parabolics

Suppose that $\stackrel{i}{x}$ and $\stackrel{i}{y}$ are incident elements in 𝒟. We require a clear understanding of the structure of the intersection $G_x^{\left[i\right]}\cap G_y^{\left[i\right]}$ in terms of the chief factors of $G_x^{\left[i\right]}$. This information, as summarized in lemmas below, is not so difficult to deduce, keeping in mind that $$ℳ=\left\{G^{\left[m\right]},G^{\left[0\right]},G^{\left[2\right]},G^{\left[3\right]}\right\}$$ is the amalgam of maximal parabolic subgroups associated with the action of G on 𝒟.

The action of $G_a^{\left[m\right]}$ on 𝒟 _a follows from the results in Sections 7.2 and 7.3, particularly from (7.2.2) and (7.3.3).

Lemma 7.7.1 Let a ∈ 𝒟^[m] and let $G_a^{\left[m\right]}\cong 2^{11}:M_{24}$ be the stabilizer of a in G. Then

1. (i) if $b\in {𝒟}_a^{\left[0\right]}$ then

   1. (1) b = (B, α), where B is an octad from ℬ _a and α ∈ {0, 1};

   (p. 144 )

   2. (2) the subgroup $M_a^{\left[m\right]}\left(b\right)\cong 2^4:L_4\left(2\right)$ stabilizes a unique hyperplane P_a (B) in $Q_a^{\left[m\right]}$;

   3. (3) $G_a^{\left[m\right]}\cap G_b^{\left[0\right]}=P_a\left(B\right):M_a^{\left[m\right]}\left(b\right)$;

   4. (4) $Q_b^{\left[0\right]}=C_{P_a\left(B\right)}\left(O_2\left(M_a^{\left[m\right]}\left(b\right)\right)\right)O_2\left(M_a^{\left[m\right]}\left(b\right)\right)\cong 2^{10}$;

2. (ii) if $c\in {𝒟}_a^{\left[2\right]}$ then

   1. (5) c is a trio from 풯 _a ;

   2. (6) the subgroup $M_a^{\left[m\right]}\left(c\right)\cong 2^6:\left(L_3\left(2\right)\times {\text{sym}}_3\right)$ stabilises in $Q_a^{\left[m\right]}$ a unique subgroup R_a (c) of index 4;

   3. (7) $G_a^{\left[m\right]}\cap G_c^{\left[2\right]}=Q_a^{\left[m\right]}:M_a^{\left[m\right]}\left(c\right)$;

   4. (8) $Q_c^{\left[2\right]}=R_a\left(c\right)O_2\left(M_a^{\left[m\right]}\left(c\right)\right)\cong 2^{3+12}$.

3. (iii) if $d\in {𝒟}_a^{\left[3\right]}$ then

   1. (9) d is a sextet from 𝒮 _a ;

   2. (10) if Y is a Sylow 3-subgroup of $O_{2,3}\left(M_a^{\left[m\right]}\left(d\right)\right)$, where $M_a^{\left[m\right]}\left(d\right)\cong 2^6:3\cdot {\text{Sym}}_6$, then $Q_a^{\left[m\right]}/\left[Q_a^{\left[m\right]},Y\right]\cong 2^5$;

   3. (11) $G_a^{\left[m\right]}\cap Q_b^{\left[3\right]}=Q_a:M_a^{\left[m\right]}\left(d\right)$;

   4. (12) $Q_d^{\left[3\right]}=\left[Q_a^{\left[m\right]},Y\right]O_2\left(M_a^{\left[m\right]}\left(d\right)\right)\cong 2_{+}^{1+12}$;

   5. (13) Y is a Sylow 3-subgroup of $O_{2,3}\left(G_d^{\left[3\right]}\right)$.

An element b of type 0 in 𝒟(G) is a vertex of the locally projective graph Γ. The edges containing b are in the natural bijection with the elements of type m incident to b in 𝒟(G). Therefore the action of $G_b^{\left[0\right]}$ on 𝒟 _b is isomorphic to the action of H ^[0] on the corresponding residue in the dual polar space 𝒪⁺(10,2) (cf. (2.1.2), (2.1.3)).

Lemma 7.7.2 Let b ∈ 𝒟^[0] and $G_b^{\left[0\right]}\cong 2^{10}:L_5\left(2\right)$ be the stabilizer of b in G.

1. (i) If $x\in {𝒟}_a^{\left[i\right]}$ for i = m, 2, or 3 then

   1. (1) x is a subspace in V_b of dimension 4, 3, or 2, respectively;

   2. (2) $G_b^{\left[0\right]}\cap G_x^{\left[i\right]}=Q_b^{\left[0\right]}:L_b^{\left[0\right]}\left(x\right)$, where $L_b^{\left[0\right]}\left(x\right)$ is isomorphic to $$2^4:L_4\left(2\right),2^6:\left(L_3\left(2\right)\times {\text{sym}}_6\right)\text{and}{2}^6:\left(L_3\left(2\right)\times {\text{sym}}_3\right)$$ in the respective three cases;

   3. (3) $Q_x^{\left[i\right]}\cap G_b^{\left[0\right]}=\left[Q_b^{\left[0\right]},O_2\left(L_b^{\left[0\right]}\left(x\right)\right)\right]O_2\left(L_b^{\left[0\right]}\left(x\right)\right)$;

   4. (4) If x is of type m then $O_2\left(G_x^{\left[m\right]}\right)$ intersects $G_b^{\left[0\right]}$ in a subgroup of index 2 in $O_2\left(G_x^{\left[m\right]}\right)$ and $Q_b^{\left[0\right]}/\left[Q_b^{\left[0\right]},O_2\left(L_b^{\left[0\right]}\left(x\right)\right)\right]\cong 2^4$;

   5. (5) the subgroup $O_2\left(G_x^{\left[i\right]}\right)$ is contained in $G_b^{\left[0\right]}$ for i = 2 and 3, while $Q_b^{\left[0\right]}/\left[Q_b^{\left[0\right]},O_2\left(L_b^{\left[0\right]}\left(x\right)\right)\right]$ is isomorphic to 2 and 2³ in the respective cases.

The next result follows from the properties of the pentad group established in Sections 4.8 and 4.9.

(p. 145 ) Lemma 7.7.3 Let c ∈ 𝒟^[2] and let $G_c^{\left[2\right]}\cong 2^{3+12}\cdot \left(L_3\left(2\right)\cdot {\text{Sym}}_5\right)$ be the stabilizer of c in G. Then

1. (i) if $a\in {𝒟}_c^{\left[m\right]}$ then

   1. (1) a is an antipodal triple in the Petersen graph Θ _c ;

   2. (2) $S_c^{\left[2\right]}\left(a\right)\cong {\text{Sym}}_4$;

   3. (3) $Q_c^{\left[2\right]}\cap G_a^{\left[m\right]}=N_c^{\left[2\right]}{S}_c^{\left[2\right]}\left(a\right)$;

   4. (4) $Q_a^{\left[m\right]}=C_{Q_c^{\left[2\right]}}\left(O_2\left(S_c^{\left[2\right]}\left(a\right)\right)\right)O_2\left(S_c^{\left[2\right]}\left(a\right)\right)$;

   5. (5) $Z_c^{\left[2\right]}\le Q_a^{\left[m\right]}$;

2. (ii) if $b\in {𝒟}_c^{\left[0\right]}$ then

   1. (6) b is a vertex of Θ _c ;

   2. (7) $S_c^{\left[2\right]}\left(b\right)\cong {\text{Sym}}_3\times 2$;

   3. (8) $G_c^{\left[2\right]}\cap G_b^{\left[0\right]}=N_c^{\left[2\right]}{S}_c^{\left[2\right]}\left(b\right)$;

   4. (9) $Q_b^{\left[0\right]}=C_{Q_c^{\left[2\right]}}\left(O_2\left(S_c^{\left[2\right]}\left(b\right)\right)\right)O_2\left(S_c^{\left[2\right]}\left(b\right)\right)$;

3. (iii) if $d\in {𝒟}_c^{\left[3\right]}$ then

   1. (10) d is a 1-dimensional subspace in $Z_c^{\left[2\right]}$;

   2. (11) $L_c^{\left[2\right]}\left(d\right)\cong {\text{Sym}}_4$;

   3. (12) $G_c^{\left[2\right]}\cap G_d^{\left[3\right]}=Q_c^{\left[2\right]}{S}_c^{\left[2\right]}{L}_c^{\left[2\right]}\left(d\right)$;

   4. (13) $Q_d^{\left[3\right]}=C_{Q_c^{\left[2\right]}}\left(O_2\left(L_c^{\left[2\right]}\left(d\right)\right)\right)O_2\left(L_c^{\left[2\right]}\left(d\right)\right)$.

The structure of $G_d^{\left[3\right]}$ and its action on 𝒟 _d follows from results in Sections 5.2, 7.3, and 7.4.

Lemma 7.7.4 Let d ∈ 𝒟^[3] and let $G_d^{\left[3\right]}\cong 2_{+}^{1+12}\cdot 3\cdot \text{Aut}\left(M_{22}\right)$ be the stabilizer of d in G. Then

1. (i) if $a\in {𝒟}_d^{\left[m\right]}$ then

   1. (1) a is a hexad from ℋ _d ;

   2. (2) $M_d^{\left[3\right]}\left(a\right)\cong 2^5:3\cdot {\text{Sym}}_6$;

   3. (3) $G_d^{\left[3\right]}\cap G_a^{\left[m\right]}=Q_d^{\left[3\right]}{M}_d^{\left[3\right]}\left(a\right)$;

   4. (4) $Q_a^{\left[m\right]}=C_{Q_d^{\left[3\right]}}\left(O_2\left(M_d^{\left[3\right]}\left(a\right)\right)\right)O_2\left(M_d^{\left[3\right]}\left(a\right)\right)$.

2. (ii) if $b\in {𝒟}_d^{\left[0\right]}$ then

   1. (5) b is an octet from 𝒪 _d ;

   2. (6) $M_d^{\left[3\right]}\left(b\right)\cong 2\times {\text{Sym}}_3\times 2^3:L_3\left(2\right)$;

   3. (8) $G_d^{\left[3\right]}\cap G_b^{\left[0\right]}=Q_d^{\left[3\right]}{M}_d^{\left[3\right]}\left(b\right)$;

   4. (9) $Q_b^{\left[0\right]}=C_{Q_d^{\left[3\right]}}\left(O_2\left(M_d^{\left[3\right]}\left(b\right)\right)\right)O_2\left(M_d^{\left[3\right]}\left(b\right)\right)\cong 2^{10}$;

3. (iii) if $c\in {𝒟}_d^{\left[2\right]}$ then

   1. (10) c is a pair from 𝒫 _d ;

   2. (11) $M_d^{\left[3\right]}\left(c\right)\cong 2^6:3:{\text{Sym}}_5$;

   (p. 146 )

   3. (12) $G_d^{\left[3\right]}\cap G_c^{\left[2\right]}=Q_d^{\left[3\right]}{M}_d^{\left[3\right]}\left(c\right)$;

   4. (13) $Q_c^{\left[2\right]}=\left[O_2\left(G_d^{\left[3\right]}\right),O_2\left(M_d^{\left[3\right]}\left(c\right)\right)\right]O_2\left(M_d^{\left[3\right]}\left(c\right)\right)\cong 2^{3+12}$.

Excerises

1. 1. Let 𝒥 = {G ^[0], G ^[1], G ^[2]} be the amalgam defined in Section 7.1. Show that the actions of N _{G ^[1]} (Q ^[m]) and N _{G ^[2]} (Q ^[m]) on Q ^[m] generate the Mathieu group M ₂₄.

2. 2. Show directly that the amalgam {F ^[m1]/Q ^[m], F ^[m2]/Q ^[m], F ^[m3]/Q ^[m]} as in (7.3.3) is isomorphic to 𝒜(M ₂₄).

3. 3. Give a computer-free proof of the simple connectedness of the rank 3 tilde geometry 𝒢(M ₂₄).