Theory of Finite Simple Groups II: Commentary on the Classification Problems (Quoted tables.pdf in DVD.1)

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) = 
			14.5.8 Conjugacy classes of N_2 = N_G(Z) = 
		14.6 Character tables of local subgroups
			14.6.1 Character table of H = 
			14.6.2 Character table of E = N_G(A) = 
			14.6.3 Character table of D = N_H(A) = 
			14.6.4 Character table of U = C_G(u) = 
			14.6.6 Character table of N_2 = N_G(Z) = 
	15. O\'Nan\'s group ON
		15.1 The centralizer of a 2-central involution
		15.2 Fusion
		15.3 3-singular classes
		15.4 Embedding Janko\'s group J_1 into ON-type groups
		15.5 Existence and uniqueness proof
		15.6 Local subgroups, fusion and character table
		15.7 Representatives of conjugacy classes
			15.7.1 Conjugacy classes of H = C_G(z) = 
			15.7.2 Conjugacy classes of D = N_H(A) = 
			15.7.3 Conjugacy classes of E = N_G(A) = 
			15.7.4 Conjugacy classes of N = N_G(3_A) = N_G(r) = 
			15.7.5 Conjugacy classes of Y = N_G(5_A) = 
			15.7.6 Conjugacy classes of N_G(7_A) \\cong R = 
			15.7.7 Conjugacy classes of Janko subgroup J = 
			15.7.8 Conjugacy classes of ON \\cong G = 
		15.8 Character tables of local subgroups
			15.8.1 Character table of H = C_G(z)
			15.8.2 Character table of D = N_H(A)
			15.8.3 Character table of E = N_G(A)
			15.8.4 Character table of N_1 = N_G(r)
			15.8.5 Character table of Y = N_G(f) = N_G(5A)
			15.8.6 Character table of R = N_G(7A)
	16. Concluding remarks and open problems
		16.1 On the monster and the baby monster
		16.2 Uniqueness problems
		16.3 Is there a 27th sporadic simple group?
		16.4 Is there a general classification scheme?
	Appendix: Table of contents of the accompanying DVD
		A.1 Folder DVD.1: Pdf files of quoted tables
		A.2 Folder DVD.2: MAGMA files of generating matrices and permutations
	References
		[1]-[16]
		[17]-[40]
		[41]-[65]
		[66]-[86]
		[87]-[107]
		[108]-[130]
		[131]-[141]
	Index
vol.2 DVD -- DVD.1: Theory of Finite Simple Groups II_ Folder: Pdf files of quoted tables  -- Gerhard O. Michler
	Contents
	1. DVD.1.1 Conway\'s simple group Co_3
		1.1 Conjugacy classes of G = \\cong Co_3
		1.2 Character table of G = \\cong Co_3
	2. DVD.1.2: Conway\'s simple group Co_2
		2.1 Conjugacy classes of G_3 = \\cong Co_2
		2.2 Character table of D(Co_2) = 
	3. DVD.1.3 Fischer\'s simple group Fi_{22}
		3.1 Conjugacy classes of G_2 = \\cong Fi_{22}
		3.2 Character table of D(Fi_{22}) = 
	4. DVD.1.4 Fischer\'s simple group Fi_{23}
		4.1 Conjugacy classes of H_2 = H(2Fi_{22}) =
		4.2 Conjugacy classes of D_2 = D(2Fi_{22}) =
		4.3 Character table of D_2 = D(2Fi_{22}) =
		4.4 Character table of H(2Fi_{22}) =
	5. DVD.1.5 Conway\'s simple group Co_1
		5.1 Conjugacy classes of D(Co_1) =
		5.2 Conjugacy classes of H_1(Co_1) =
		5.3 Conjugacy classes of U(Co_1) =
		5.4 Conjugacy classes of C_{Co_1}(2b)
		5.5 Conjugacy classes of C_{Co_1}(2c) = 
		5.6 Character table of D_{Co_1} = 
		5.7 Character table of H_1(Co_1) =
		5.8 Character table of U(Co_1) =
		5.9 Character table of C_{Co_1}(2b)
		5.10 Character table of C_{Co_1}(2c)
	6. DVD.1.6 Janko\'s simple group J_4
		6.1 Conjugacy classes of D(J_4) = 
		6.2 Conjugacy classes of C_{J_4}(2b)
		6.3 Character table of D = D(J_4) = 
		6.4 Character table of C_{J_4}(2b)
	7. DVD.1.7 Fischer\'s simple group Fi\'_{24}
		7.1 Conjugacy classes of D(Fi\'_{24}) = 
		7.2 Character table of mH = 
		7.3 Character table of mE = 
		7.4 Character table of D(Fi\'_{24}) =