The fourth Janko group 1st edition by Ivanov 9780191523625 0191523623

1 Concrete group theory

1.1 Symplectic and orthogonal GF(2)-forms

1.2 S[sub(2m)](2), O[sup(+)][sub(2m)](2), and O[sup(–)][sub(2m)](2)

1.3 Transvections and Siegel transformations

1.4 Exterior square via symplectic forms

1.5 Witt’s theorem

1.6 Extraspecial 2-groups

1.7 The space of forms

1.8 S[sub(4)](2) and Sym[sub(6)]

1.9 Ω[sup(+)][sub(6)](2), Alt[sub(8)], and L[sub(4)](2)

1.10 L[sub(2)](7) and L[sub(3)](2)

2 O[sup(+)][sub(10)](2) as a prototype

2.1 Dual polar graph

2.2 Geometric cubic subgraphs

2.3 Amalgam H = {H[sup([0])], H[sup([1])]}

2.4 The universal completion

2.5 Characterization

3 Modifying the rank 2 amalgam

3.1 Complements and first cohomology

3.2 Permutation modules

3.3 Deck automorphisms of 2[sup(6)] : L[sub(4)](2)

3.4 Automorphism group of H[sup([01])]

3.5 Amalgam G = {G[sup([0])],G[sup([1])]}

3.6 Vectors and hyperplanes in Q[sup([0])]

3.7 Structure of G[sup([1])]

3.8 A shade of a Mathieu group

4 Pentad group 2[sup(3+12)] · (L[sub(3)](2) × Sym[sub(5)])

4.1 Geometric subgroups and subgraphs

4.2 Kernels and actions

4.3 Inspecting N[sup([2])]

4.4 Cohomology of L[sub(3)](2)

4.5 Trident group

4.6 Automorphism group of N[sup([2])]

4.7 {D[sub(12)], D[sub(8)]}-amalgams in Sym[sub(6)]

4.8 The last inch

4.9 Some properties of the pentad group

5 Towards 2[sup(1+12)][sub(+)] · 3 · Aut (M[sub(22)])

5.1 Automorphism group of N[sup([3])]

5.2 A Petersen-type amalgam in G[sup([3])

5.3 The 12-dimensional module

5.4 The triviality of N[sup([4])]

6 The 1333-dimensional representation

6.1 Representations of rank 2 amalgams

6.2 Bounding the dimension

6.3 On irreducibility of induced modules

6.4 Representing G[sup([0])]

6.5 Restricting to G[sup([01])]

6.6 Lifting Π[sup([01])][sub(11)]

6.7 Lifting Π[sup([01])][sub(22)]

6.8 Lifting Π[sup([01])][sub(12)] ⊕ Π[sup([01])][sub(21)]

6.8.1 24-dim representations of C[sub(G[1])] (h)

6.8.2 Structure of Π[sup((h))][sub(12)]

6.8.3 Structure of Π[sup((h))][sub(21)]

6.8.4 Gluing

6.9 The minimal representations of G

6.10 The action of G[sup([2])]

6.11 The centralizer of N[sup([2])]

6.12 The fundamental group of the Petersen graph

6.13 Completion constrained at level 2

7 Getting the parabolics together

7.1 Encircling 2[sup(1+12)][sub(+)] · 3 · Aut (M[sub(22)])

7.2 Tracking 2[sup(11)] : M[sub(24)]

7.3 P-geometry of G[sup([4])]

7.4 G[sup([4]) = G

7.5 Maximal parabolic geometry D

7.6 Residues in D

7.7 Intersections of maximal parabolics

8 173,067,389-vertex graph Δ

8.1 Defining the graph

8.2 The local graph of Δ

8.3 Distance two neighbourhood

8.3.1 Analysing 2-paths

8.3.2 Calculating μ2

8.3.3 Calculating μ3

8.3.4 The μ-subgraphs

8.4 Earthing up Δ[sup(1)][sub(3)](a)

8.5 Earthing up Δ[sup(2)][sub(3)](a)

8.6 | G | = 2[sup(21)] · 3[sup(3)] · 5 · 7 · 11[sup(3)] · 23 · 29 · 31 · 37 · 43

8.7 The simplicity of G

8.8 The involution centralizer

9 History and beyond

9.1 Janko’s discovery

9.2 Characterizations

9.3 Ronan–Smith geometry

9.4 Cambridge five

9.5 P-geometry

9.6 Uniqueness of J[sub(4)]

9.7 Lempken’s construction

9.8 Computer-free construction

9.9 The maximal subgroups

9.10 Rowley–Walker diagram

9.11 Locally projective graphs

9.12 On the 112-dimensional module

9.13 Miscellaneous

10 Appendix: Terminology and notation

10.1 Groups

10.2 Amalgams

10.3 Graphs

10.4 Diagram geometries

11 Appendix: Mathieu groups and their geometries

11.1 Witt design S(5, 8, 24)

11.2 Geometries of M[sub(24)]

11.3 Golay code and Todd modules

11.4 Shpectorov’s characterization of M[sub(22)]

11.5 Diagrams of H(M[sub(24)])

11.6 Diagrams of H(M[sub(22)])

References

Index