Theory Of Finite Simple Groups Ii: Commentary On The Classification Problems Theory Of Finite Simple Groups 2 Theory Of Finite Simple Groups Two

This is a coherent explanation for the existence of the 26 known sporadic simple groups originally arising from many unrelated contexts. The given proofs build on the close relations between general group theory, ordinary character theory, modular representation theory and algorithmic algebra described in the first volume. The author presents a new algorithm by which 25 sporadic simple groups can be constructed (the smallest Mathieu group M11 can be omitted for theoretical reasons), and demonstrates that it is not restricted to sporadic simple groups. He also describes the constructions of various groups and proves their uniqueness whenever possible. The computational existence proofs are documented in the accompanying DVD. The author also states several open problems related to the theorem asserting that there are exactly 26 groups, and R. Brauer's warning that there may be infinitely many. Some of these problems require new experiments with the author's algorithm. GERHARD MICHLER, Theory of Finite Simple Groups II -- Commentary on the Classification Problems Contents Acknowledgements Introduction 1. Simple groups and indecomposable subgroups of GL_n(2) 1.1 Two alternative views on the classification problem 1.2 Simple groups are of infinite representation type, p = 2 1.3 The algorithm 1.4 Documentation of experimental results 1.5 Constructing projective irreducible modular representations 1.6 Thompson's group order formula revisited 2. Dickson group G_2(3) and related simple groups 2.1 Involution centralizers of Dickson's groups G_2(q), q odd 2.2 Fusion and conjugacy classes of even order 2.3 The 3-singular conjugacy classes 2.4 Janko's characterization of G_2(3) 2.5 Representatives of conjugacy classes 2.5.1 Conjugacy classes of W= 2.5.2 Conjugacy classes of X_1= 2.5.3 Conjugacy classes of X_2= 2.5.4 Conjugacy classes of N_G(d_1) = X_3 = 2.6 Character tables of local subgroups of G_2(3) 2.6.1 Character table of N_G(3_A) \cong N_G(3_B) 2.6.2 Character table of N_G(3_D) \cong N_G(3_E) 2.6.3 Character table of N_G(7_A) 2.6.4 Character table of N_G(13_A) \cong N_G(13_B) 3. Conway's simple group Co_3 3.1 Construction of the involution centralizer 3.2 Construction of a simple group of Co_3-type 3.3 Uniqueness proof 3.4 Representatives of conjugacy classes 3.4.1 Conjugacy classes of H= 3.4.2 Conjugacy classes of E= 3.4.3 Conjugacy classes of D= 3.4.4 Conjugacy classes of N_1 = N_G(r_1) = 3.4.5 Conjugacy classes of N_2 = N_G(r_2) = 3.4.6 Conjugacy classes of N_3 = N_G(r_3) = 3.4.7 Conjugacy classes of N_5 = N_G(f_1) = 3.4.8 Conjugacy classes of N_6 = N_G(f_2) = 3.5 Character tables of local subgroups 3.5.1 Character table of E= 3.5.2 Character table of D= 3.5.3 Character table of H= 3.5.4 Character table of U=C_G(u) \cong \times Q 3.5.5 Character table of N_1 = N_G(r_1) = 3.5.6 Character table of N_2 = N_G(r_2) = 3.5.7 Character table of N_3 = N_G(r_3) = 3.5.8 Character table of N_5 = N_G(f_1) = 3.5.9 Character table of N_6 = N_G(f_2) = 4. Conway's simple group Co_2 4.1 Extensions of the Mathieu group M_{22} and Aut(M_{22}) 4.2 Construction of the 2-central involution centralizer 4.3 Construction of Conway's simple group Co_2 4.4 On the uniqueness of Co_2 4.5 Representatives of conjugacy classes 4.5.1 Conjugacy classes of H(Co_2) = 4.5.2 Conjugacy classes of D(Co_2) = 4.5.3 Conjugacy classes of E(Co_2) = 4.6 Character tables of local subgroups of Co_2 4.6.1 Character table of E_3 = E(Co_2) = 4.6.2 Character table of H(Co_2) = 5. Fischer's simple group Fi_{22} 5.1 Construction of the 2-central involution centralizers 5.2 Construction of Fischer's simple group Fi_{22} 5.3 Sketch of a uniqueness proof 5.4 The remaining cases E_1, E_4 and E_5 5.5 Representatives of conjugacy classes 5.5.1 Conjugacy classes of H(Fi_{22}) = 5.5.2 Conjugacy classes of D(Fi_{22}) = 5.5.3 Conjugacy classes of E(Fi_{22}) = 5.6 Character tables of local subgroups of Fi_{22} 5.6.1 Character table of E_2 = E(Fi_{22}) = 5.6.2 Character table of H(Fi_{22}) = 6. Fischer's simple group Fi_{23} 6.1 Extensions of the Mathieu group M_{23} 6.2 Construction of a 2-central involution centralizer 6.3 Construction of Fischer's simple group Fi_{23} 6.4 On the uniqueness of Fi_{23} 6.5 Representatives of conjugacy classes 6.5.1 Conjugacy classes of E(Fi_{23}) = 6.5.2 Conjugacy classes of H(Fi_{23}) = 6.5.3 Conjugacy classes of D(Fi_{23}) = 6.5.4 Conjugacy classes of Fi_{23} = 6.6 Character tables of local subgroups of Fi_{23} 6.6.1 Character table of E = E(Fi_{23}) = 6.6.2 Character table of D(Fi_{23}) = 6.6.3 Character table of H(Fi_{23}) = 7. Conway's simple group Co_1 7.1 Extensions of the Mathieu group M_{24} 7.2 Construction of the 2-central involution centralizer of Co_1 7.3 Construction of Conway's simple group Co_1 7.4 On the uniqueness of Co_1 7.5 Representatives of conjugacy classes 7.5.1 Conjugacy classes of E(Co_1) = 7.5.2 Conjugacy classes of H(Co_1) = 7.6 Character tables of local subgroups of Co_1 7.6.1 Character table of E(Co_1) = 7.6.2 Character table of H(Co_1) = 8. Janko's group J_4 8.1 Structure of the given centralizer 8.2 Conjugacy classes and group order 8.3 Existence and uniqueness proofs 8.4 Other constructions in GL_{1333}(11) and GL_{112}(2) 8.5 Representatives of conjugacy classes 8.5.1 Conjugacy classes of H = C_G(z) = 8.5.2 Conjugacy classes of E = N_G(A) = 8.6 Character tables of local subgroups 8.6.1 Character table of H(J_4) = 8.6.2 Character table of E = N_G(A) = 9. Fischer's simple group Fi'_{24} 9.1 The 2-fold cover of the automorphism group Aut(Fi_{22}) 9.2 A semi-simple representation of Fi_{23} in GL_{8671}(13) 9.3 Construction of the irreducible subgroup G of GL_{8671}(13) 9.4 G is isomorphic to Fischer's simple group Fi'_{24} 9.5 Presentation of 2-central involution centralizer 9.6 On the uniqueness of Fi'_{24} 9.7 Representatives of conjugacy classes 9.7.1 Conjugacy classes of A_1 = 2Aut(Fi_{22}) = 9.7.2 Conjugacy classes of E(Fi_{24}) = 9.7.3 Conjugacy classes of H(Fi'_{24}) = 9.8 Character tables of local subgroups 9.8.1 Character table of A_1 = 2Aut(Fi_{22}) = 9.8.2 Character table of H(Fi'_{24}) = 9.8.3 Character table of E(Fi'_{24}) = 10. Tits' group ^2F_4(2)' 10.1 Construction of the 2-central involution centralizer 10.2 Fusion 10.3 Existence proof of Tits' simple group inside GL_{26}(73) 10.4 Group order 10.5 The 3-, 5- and 13-singular conjugacy classes 10.6 Uniqueness proof 10.7 Representatives of conjugacy classes 10.7.1 Conjugacy classes of H = 10.7.2 Conjugacy classes of N_G(S_5) = 10.7.3 Conjugacy classes of D = 10.7.4 Conjugacy classes of E = 10.7.5 Conjugacy classes of U = 10.7.6 Conjugacy classes of N_3 = 10.7.7 Conjugacy classes of N_5 = 10.8 Character tables of local subgroups 10.8.1 Character table of H = C_G(z) 10.8.2 Character table of D = N_H(A) 10.8.3 Character table of E = N_G(A) 10.8.4 Character table of U = C_G(u) 10.8.5 Character table of N_3 10.8.6 Character table of N_5 10.8.7 Character table of NS_5 = N_G(S_5) 11. McLaughlin's group McL 11.1 Construction of the 2-central involution centralizer 11.2 Structure of the given centralizer H = 2A_8 11.3 Existence and uniqueness proof 11.4 Representatives of conjugacy classes 11.4.1 Conjugacy classes of E = 11.4.2 Conjugacy classes of H = 11.4.3 Conjugacy classes of D = 11.4.4 Conjugacy classes of G = 11.5 Character tables of local subgroups 11.5.1 Character table of E = 11.5.2 Character table of D = 11.5.3 Character table of H = 12. Rudvalis' group Ru 12.1 Construction of the 2-central involution centralizer 12.2 Construction of a simple group of Ru-type 12.3 Fusion 12.4 Uniqueness proof 12.5 Representatives of conjugacy classes 12.5.1 Conjugacy classes of H = 12.5.2 Conjugacy classes of D = 12.5.3 Conjugacy classes of E = 12.5.4 Conjugacy classes of M = 12.5.5 Conjugacy classes of G = 12.6 Character tables of local subgroups 12.6.1 Character table of H = 12.6.2 Character table of E= 12.6.3 Character table of D= 12.6.4 Character table of N_G(3_A) \cong 3Aut(A_6) 12.6.5 Character table of M = N_G(R) = (d_1i, d_2) 12.6.6 Character table of N_G(5_A) \cong 5^{1+2} : (Q_8 \times 4) 12.6.7 Character table of N_G(5_B) \cong 5 : 4 \times A_5 12.6.8 Character table of G= 13. Lyons' group Ly 13.1 Structure of the given centralizer 13.2 Conjugacy classes of elements of even order 13.3 Conjugacy classes of p-singular elements 13.4 Group order 13.5 Existence and uniqueness proofs 13.6 Representatives of conjugacy classes 13.6.1 Conjugacy classes of H = 13.6.2 Conjugacy classes of D = N_H(A) = 13.6.3 Conjugacy classes of N = N_G (A)= 13.6.4 Conjugacy classes of E = N_G(3_A) = \cong 3McL : 2 13,6.5 Conjugacy classes of R = N_G(f) = 13.6.6 Conjugacy classes of L = N_R(V) = 13.6. 7 Conjugacy classes of M = N_G(V) = 13.7 Character tables of local subgroups 13.7.1 Character table of H = \cong 2A_{11} 13.7.2 Character table of D = 13.7.3 Character table of N = 13.7.4 Character table of E = N_G(3_A) \cong 3McL : 2 13.7.5 Character table of R = N_G(f) \cong 5^{1+4} : 4S_6 13.7.6 Character table of M = N_G(V) \cong 5^3.L_3(5) 13.7.7 Character table of L = N_R(V) \cong 5^3. (5^2 : GL_2(5)) 14. Suzuki's group Suz 14.1 The centralizer of a 2-central involution 14.2 Even conjugacy classes and group order 14.3 Existence proof of Suz inside GL_{143}(13) 14.4 Uniqueness proof 14.5 Representatives of conjugacy classes 14.5.1 Conjugacy classes of H = 14.5.2 Conjugacy classes of E = N_G(A) = 14.5.3 Conjugacy classes of D = 14.5.4 Conjugacy classes of W = 14.5.5 Conjugacy classes of M = N_G(V) = 14.5.6 Conjugacy classes of C_G(u) =