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Generation of finite groups with applications to computing normalisers Colva Roney-Dougal University of St Andrews 11th October 2017 Colva Roney-Dougal University of St Andrews Generation of finite groups Minimal generation Colva

  1. Probabilistic generation of simple groups P G ( k ) – probability that k random elts of G generate G . Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then P S (2) → 1 as | S | → ∞ . Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then P S (2) ≥ 53 / 90 , with equality if and only if S = A 6 . m ( G ) – minimal index of a proper subgroup of G . Theorem (Liebeck & Shalev 96) There exist constants α and β s.t. for all finite simple groups S, Colva Roney-Dougal University of St Andrews Generation of finite groups

  2. Probabilistic generation of simple groups P G ( k ) – probability that k random elts of G generate G . Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then P S (2) → 1 as | S | → ∞ . Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then P S (2) ≥ 53 / 90 , with equality if and only if S = A 6 . m ( G ) – minimal index of a proper subgroup of G . Theorem (Liebeck & Shalev 96) There exist constants α and β s.t. for all finite simple groups S, α β 1 − m ( S ) < P S (2) < 1 − m ( S ) . Colva Roney-Dougal University of St Andrews Generation of finite groups

  3. Probabilistic generation of simple groups P G ( k ) – probability that k random elts of G generate G . Theorem (Dixon 69; Kantor & Lubotzky 90; Liebeck & Shalev 95) S – finite simple group. Then P S (2) → 1 as | S | → ∞ . Theorem (Menezes, Quick, CMRD 13) S – finite simple group. Then P S (2) ≥ 53 / 90 , with equality if and only if S = A 6 . m ( G ) – minimal index of a proper subgroup of G . Theorem (Liebeck & Shalev 96) There exist constants α and β s.t. for all finite simple groups S, α β 1 − m ( S ) < P S (2) < 1 − m ( S ) . Colva Roney-Dougal University of St Andrews Generation of finite groups

  4. Minimal generation of permutation groups Colva Roney-Dougal University of St Andrews Generation of finite groups

  5. Minimal generation of permutation groups G ≤ S n is transitive Colva Roney-Dougal University of St Andrews Generation of finite groups

  6. Minimal generation of permutation groups G ≤ S n is transitive if for all α, β ∈ { 1 , . . . , n } there exists g ∈ G s.t. α g = β . Colva Roney-Dougal University of St Andrews Generation of finite groups

  7. Minimal generation of permutation groups G ≤ S n is transitive if for all α, β ∈ { 1 , . . . , n } there exists g ∈ G s.t. α g = β . Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ S n . Then d ( G ) ≤ max { n / 2 , 2 } . Colva Roney-Dougal University of St Andrews Generation of finite groups

  8. Minimal generation of permutation groups G ≤ S n is transitive if for all α, β ∈ { 1 , . . . , n } there exists g ∈ G s.t. α g = β . Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ S n . Then d ( G ) ≤ max { n / 2 , 2 } . Bound is best possible: If n is even then C n / 2 ≤ S n , and d ( C n / 2 ) = n / 2. 2 2 Colva Roney-Dougal University of St Andrews Generation of finite groups

  9. Minimal generation of permutation groups G ≤ S n is transitive if for all α, β ∈ { 1 , . . . , n } there exists g ∈ G s.t. α g = β . Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ S n . Then d ( G ) ≤ max { n / 2 , 2 } . Bound is best possible: If n is even then C n / 2 ≤ S n , and d ( C n / 2 ) = n / 2. 2 2 d (S 3 ) = 2. Key ingredient of proof is: Lemma (Wielandt) Let P ≤ S p m be a transitive p-group. Colva Roney-Dougal University of St Andrews Generation of finite groups

  10. Minimal generation of permutation groups G ≤ S n is transitive if for all α, β ∈ { 1 , . . . , n } there exists g ∈ G s.t. α g = β . Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ S n . Then d ( G ) ≤ max { n / 2 , 2 } . Bound is best possible: If n is even then C n / 2 ≤ S n , and d ( C n / 2 ) = n / 2. 2 2 d (S 3 ) = 2. Key ingredient of proof is: Lemma (Wielandt) Let P ≤ S p m be a transitive p-group. Then d ( P ) ≤ 1 + � m − 2 i =0 p i . Colva Roney-Dougal University of St Andrews Generation of finite groups

  11. Minimal generation of permutation groups G ≤ S n is transitive if for all α, β ∈ { 1 , . . . , n } there exists g ∈ G s.t. α g = β . Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ S n . Then d ( G ) ≤ max { n / 2 , 2 } . Bound is best possible: If n is even then C n / 2 ≤ S n , and d ( C n / 2 ) = n / 2. 2 2 d (S 3 ) = 2. Key ingredient of proof is: Lemma (Wielandt) Let P ≤ S p m be a transitive p-group. Then d ( P ) ≤ 1 + � m − 2 i =0 p i . Corollary If P ≤ S n is a p-group, then d ( P ) ≤ n / 2 . Colva Roney-Dougal University of St Andrews Generation of finite groups

  12. Minimal generation of permutation groups G ≤ S n is transitive if for all α, β ∈ { 1 , . . . , n } there exists g ∈ G s.t. α g = β . Theorem (Cameron, Solomon & Turull 89; Neumann) Let G ≤ S n . Then d ( G ) ≤ max { n / 2 , 2 } . Bound is best possible: If n is even then C n / 2 ≤ S n , and d ( C n / 2 ) = n / 2. 2 2 d (S 3 ) = 2. Key ingredient of proof is: Lemma (Wielandt) Let P ≤ S p m be a transitive p-group. Then d ( P ) ≤ 1 + � m − 2 i =0 p i . Corollary If P ≤ S n is a p-group, then d ( P ) ≤ n / 2 . Colva Roney-Dougal University of St Andrews Generation of finite groups

  13. Minimal generation of transitive groups Colva Roney-Dougal University of St Andrews Generation of finite groups

  14. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive Colva Roney-Dougal University of St Andrews Generation of finite groups

  15. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 Colva Roney-Dougal University of St Andrews Generation of finite groups

  16. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 and ( G , n ) � = ( D 8 ◦ D 8 , 8) Colva Roney-Dougal University of St Andrews Generation of finite groups

  17. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 and ( G , n ) � = ( D 8 ◦ D 8 , 8) then d ( G ) < n / 2 . Colva Roney-Dougal University of St Andrews Generation of finite groups

  18. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 and ( G , n ) � = ( D 8 ◦ D 8 , 8) then d ( G ) < n / 2 . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ S n is transitive Colva Roney-Dougal University of St Andrews Generation of finite groups

  19. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 and ( G , n ) � = ( D 8 ◦ D 8 , 8) then d ( G ) < n / 2 . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ S n is transitive, then d ( G ) ≤ cn / √ log n. Colva Roney-Dougal University of St Andrews Generation of finite groups

  20. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 and ( G , n ) � = ( D 8 ◦ D 8 , 8) then d ( G ) < n / 2 . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ S n is transitive, then d ( G ) ≤ cn / √ log n. Kovacs and Newman: for each prime p there exists a constant c p Colva Roney-Dougal University of St Andrews Generation of finite groups

  21. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 and ( G , n ) � = ( D 8 ◦ D 8 , 8) then d ( G ) < n / 2 . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ S n is transitive, then d ( G ) ≤ cn / √ log n. Kovacs and Newman: for each prime p there exists a constant c p s.t. for all b there exists a transitive p -subgroup P ≤ S p b = S n Colva Roney-Dougal University of St Andrews Generation of finite groups

  22. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 and ( G , n ) � = ( D 8 ◦ D 8 , 8) then d ( G ) < n / 2 . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ S n is transitive, then d ( G ) ≤ cn / √ log n. Kovacs and Newman: for each prime p there exists a constant c p s.t. for all b there exists a transitive p -subgroup P ≤ S p b = S n with d ( P ) > c p n / √ log n . Colva Roney-Dougal University of St Andrews Generation of finite groups

  23. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 and ( G , n ) � = ( D 8 ◦ D 8 , 8) then d ( G ) < n / 2 . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ S n is transitive, then d ( G ) ≤ cn / √ log n. Kovacs and Newman: for each prime p there exists a constant c p s.t. for all b there exists a transitive p -subgroup P ≤ S p b = S n with d ( P ) > c p n / √ log n . Theorem (Tracey 17) √ Can take c = 0 . 92 , or 3 / 2 with finitely many exceptions. (All logs to base 2, unless otherwise stated.) Colva Roney-Dougal University of St Andrews Generation of finite groups

  24. Minimal generation of transitive groups Theorem (Cameron Solomon Turull; Neumann 89) If G ≤ S n is transitive, n > 4 and ( G , n ) � = ( D 8 ◦ D 8 , 8) then d ( G ) < n / 2 . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant c s.t. if G ≤ S n is transitive, then d ( G ) ≤ cn / √ log n. Kovacs and Newman: for each prime p there exists a constant c p s.t. for all b there exists a transitive p -subgroup P ≤ S p b = S n with d ( P ) > c p n / √ log n . Theorem (Tracey 17) √ Can take c = 0 . 92 , or 3 / 2 with finitely many exceptions. (All logs to base 2, unless otherwise stated.) Colva Roney-Dougal University of St Andrews Generation of finite groups

  25. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . Colva Roney-Dougal University of St Andrews Generation of finite groups

  26. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ Colva Roney-Dougal University of St Andrews Generation of finite groups

  27. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ Colva Roney-Dougal University of St Andrews Generation of finite groups

  28. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . Colva Roney-Dougal University of St Andrews Generation of finite groups

  29. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive Colva Roney-Dougal University of St Andrews Generation of finite groups

  30. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive and all blocks have size 1 or n . Colva Roney-Dougal University of St Andrews Generation of finite groups

  31. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive and all blocks have size 1 or n . Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c Colva Roney-Dougal University of St Andrews Generation of finite groups

  32. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive and all blocks have size 1 or n . Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ S n is primitive Colva Roney-Dougal University of St Andrews Generation of finite groups

  33. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive and all blocks have size 1 or n . Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ S n is primitive then c log n d ( G ) ≤ √ log log n . Colva Roney-Dougal University of St Andrews Generation of finite groups

  34. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive and all blocks have size 1 or n . Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ S n is primitive then c log n d ( G ) ≤ √ log log n . Theorem (Holt & CMRD 12) Let G ≤ S n be a subnormal subgroup of a primitive group. Colva Roney-Dougal University of St Andrews Generation of finite groups

  35. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive and all blocks have size 1 or n . Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ S n is primitive then c log n d ( G ) ≤ √ log log n . Theorem (Holt & CMRD 12) Let G ≤ S n be a subnormal subgroup of a primitive group. Then d ( G ) ≤ max { log n , 2 } . Colva Roney-Dougal University of St Andrews Generation of finite groups

  36. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive and all blocks have size 1 or n . Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ S n is primitive then c log n d ( G ) ≤ √ log log n . Theorem (Holt & CMRD 12) Let G ≤ S n be a subnormal subgroup of a primitive group. Then d ( G ) ≤ max { log n , 2 } . Bound is best possible: Colva Roney-Dougal University of St Andrews Generation of finite groups

  37. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive and all blocks have size 1 or n . Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ S n is primitive then c log n d ( G ) ≤ √ log log n . Theorem (Holt & CMRD 12) Let G ≤ S n be a subnormal subgroup of a primitive group. Then d ( G ) ≤ max { log n , 2 } . Bound is best possible: Consider K = ( F m 2 , +) ✂ AGL m (2) ≤ S 2 m . Colva Roney-Dougal University of St Andrews Generation of finite groups

  38. Minimal generation of primitive groups Let ∆ ⊆ { 1 , . . . , n } . If for all g ∈ G , either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ , then ∆ is a block for G . G ≤ S n is primitive if G is transitive and all blocks have size 1 or n . Theorem (Lucchini, Menegazzo & Morigi 01) There exists a constant c such that if G ≤ S n is primitive then c log n d ( G ) ≤ √ log log n . Theorem (Holt & CMRD 12) Let G ≤ S n be a subnormal subgroup of a primitive group. Then d ( G ) ≤ max { log n , 2 } . Bound is best possible: Consider K = ( F m 2 , +) ✂ AGL m (2) ≤ S 2 m . Colva Roney-Dougal University of St Andrews Generation of finite groups

  39. Random generation of permutation groups Colva Roney-Dougal University of St Andrews Generation of finite groups

  40. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Colva Roney-Dougal University of St Andrews Generation of finite groups

  41. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Colva Roney-Dougal University of St Andrews Generation of finite groups

  42. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Let ǫ ∈ (0 , 1) Colva Roney-Dougal University of St Andrews Generation of finite groups

  43. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Let ǫ ∈ (0 , 1) , and let t be such that ζ ( t ) ≤ 1 + ǫ . Colva Roney-Dougal University of St Andrews Generation of finite groups

  44. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Let ǫ ∈ (0 , 1) , and let t be such that ζ ( t ) ≤ 1 + ǫ . Let G ≤ S n . Colva Roney-Dougal University of St Andrews Generation of finite groups

  45. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Let ǫ ∈ (0 , 1) , and let t be such that ζ ( t ) ≤ 1 + ǫ . Let G ≤ S n . Then d ǫ ( G ) < n / 2 + 2(log n + log log n ) + t + 2 . Colva Roney-Dougal University of St Andrews Generation of finite groups

  46. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Let ǫ ∈ (0 , 1) , and let t be such that ζ ( t ) ≤ 1 + ǫ . Let G ≤ S n . Then d ǫ ( G ) < n / 2 + 2(log n + log log n ) + t + 2 . If G is transitive then d ǫ ( G ) < 0 . 92 n √ log n + 2(log n + log log n ) + t + 2 . Colva Roney-Dougal University of St Andrews Generation of finite groups

  47. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Let ǫ ∈ (0 , 1) , and let t be such that ζ ( t ) ≤ 1 + ǫ . Let G ≤ S n . Then d ǫ ( G ) < n / 2 + 2(log n + log log n ) + t + 2 . If G is transitive then d ǫ ( G ) < 0 . 92 n √ log n + 2(log n + log log n ) + t + 2 . If G is a subnormal subgroup of a primitive group Colva Roney-Dougal University of St Andrews Generation of finite groups

  48. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Let ǫ ∈ (0 , 1) , and let t be such that ζ ( t ) ≤ 1 + ǫ . Let G ≤ S n . Then d ǫ ( G ) < n / 2 + 2(log n + log log n ) + t + 2 . If G is transitive then d ǫ ( G ) < 0 . 92 n √ log n + 2(log n + log log n ) + t + 2 . If G is a subnormal subgroup of a primitive group, then d ǫ ( G ) < 3 log n + 2 log log n + t + 2 . Colva Roney-Dougal University of St Andrews Generation of finite groups

  49. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Let ǫ ∈ (0 , 1) , and let t be such that ζ ( t ) ≤ 1 + ǫ . Let G ≤ S n . Then d ǫ ( G ) < n / 2 + 2(log n + log log n ) + t + 2 . If G is transitive then d ǫ ( G ) < 0 . 92 n √ log n + 2(log n + log log n ) + t + 2 . If G is a subnormal subgroup of a primitive group, then d ǫ ( G ) < 3 log n + 2 log log n + t + 2 . If G is primitive then d ǫ ( G ) < log n + log log n + t + 4 . 59 . Colva Roney-Dougal University of St Andrews Generation of finite groups

  50. Random generation of permutation groups Let d ǫ ( G ) be the minimum number of random elements needed to generate G with probability at least 1 − ǫ . Theorem (Various people) Let ǫ ∈ (0 , 1) , and let t be such that ζ ( t ) ≤ 1 + ǫ . Let G ≤ S n . Then d ǫ ( G ) < n / 2 + 2(log n + log log n ) + t + 2 . If G is transitive then d ǫ ( G ) < 0 . 92 n √ log n + 2(log n + log log n ) + t + 2 . If G is a subnormal subgroup of a primitive group, then d ǫ ( G ) < 3 log n + 2 log log n + t + 2 . If G is primitive then d ǫ ( G ) < log n + log log n + t + 4 . 59 . Colva Roney-Dougal University of St Andrews Generation of finite groups

  51. What is a random subgroup of S n ? Colva Roney-Dougal University of St Andrews Generation of finite groups

  52. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Colva Roney-Dougal University of St Andrews Generation of finite groups

  53. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? Colva Roney-Dougal University of St Andrews Generation of finite groups

  54. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n Colva Roney-Dougal University of St Andrews Generation of finite groups

  55. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Colva Roney-Dougal University of St Andrews Generation of finite groups

  56. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Uniform random amongst the subgroups of S n ? Colva Roney-Dougal University of St Andrews Generation of finite groups

  57. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Uniform random amongst the subgroups of S n ? Uniform random amongst the conjugacy classes of subgroups? Colva Roney-Dougal University of St Andrews Generation of finite groups

  58. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Uniform random amongst the subgroups of S n ? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a ( n ) – number of subgroups of S n . Colva Roney-Dougal University of St Andrews Generation of finite groups

  59. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Uniform random amongst the subgroups of S n ? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a ( n ) – number of subgroups of S n . Then 2 n 2 (1 / 16+ o (1)) ≤ a ( n ) ≤ 2 n 2 (log 2 (24) / 6+ o (1)) . Colva Roney-Dougal University of St Andrews Generation of finite groups

  60. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Uniform random amongst the subgroups of S n ? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a ( n ) – number of subgroups of S n . Then 2 n 2 (1 / 16+ o (1)) ≤ a ( n ) ≤ 2 n 2 (log 2 (24) / 6+ o (1)) . Lower bound: Colva Roney-Dougal University of St Andrews Generation of finite groups

  61. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Uniform random amongst the subgroups of S n ? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a ( n ) – number of subgroups of S n . Then 2 n 2 (1 / 16+ o (1)) ≤ a ( n ) ≤ 2 n 2 (log 2 (24) / 6+ o (1)) . Lower bound: consider C ⌊ n / 2 ⌋ = F ⌊ n / 2 ⌋ ∼ < S n 2 2 Colva Roney-Dougal University of St Andrews Generation of finite groups

  62. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Uniform random amongst the subgroups of S n ? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a ( n ) – number of subgroups of S n . Then 2 n 2 (1 / 16+ o (1)) ≤ a ( n ) ≤ 2 n 2 (log 2 (24) / 6+ o (1)) . Lower bound: consider C ⌊ n / 2 ⌋ = F ⌊ n / 2 ⌋ ∼ < S n , and count subspaces. 2 2 Colva Roney-Dougal University of St Andrews Generation of finite groups

  63. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Uniform random amongst the subgroups of S n ? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a ( n ) – number of subgroups of S n . Then 2 n 2 (1 / 16+ o (1)) ≤ a ( n ) ≤ 2 n 2 (log 2 (24) / 6+ o (1)) . Lower bound: consider C ⌊ n / 2 ⌋ = F ⌊ n / 2 ⌋ ∼ < S n , and count subspaces. 2 2 Hence: not much difference between “random amongst subgroups” and “random amongst conjugacy classes of subgroups”. Colva Roney-Dougal University of St Andrews Generation of finite groups

  64. What is a random subgroup of S n ? More than one possible interpretation of “random subgroup”: Uniformly randomly generated? With probability tending rapidly to 1, this is S n (or cyclic, or A n ). Uniform random amongst the subgroups of S n ? Uniform random amongst the conjugacy classes of subgroups? Theorem (Pyber 93) a ( n ) – number of subgroups of S n . Then 2 n 2 (1 / 16+ o (1)) ≤ a ( n ) ≤ 2 n 2 (log 2 (24) / 6+ o (1)) . Lower bound: consider C ⌊ n / 2 ⌋ = F ⌊ n / 2 ⌋ ∼ < S n , and count subspaces. 2 2 Hence: not much difference between “random amongst subgroups” and “random amongst conjugacy classes of subgroups”. Colva Roney-Dougal University of St Andrews Generation of finite groups

  65. More on random subgroups of S n Colva Roney-Dougal University of St Andrews Generation of finite groups

  66. More on random subgroups of S n P – property of permutation groups. Colva Roney-Dougal University of St Andrews Generation of finite groups

  67. More on random subgroups of S n P – property of permutation groups. If have a bound f P ( n ) on the number of generators of a subgroup of S n with property P Colva Roney-Dougal University of St Andrews Generation of finite groups

  68. More on random subgroups of S n P – property of permutation groups. If have a bound f P ( n ) on the number of generators of a subgroup of S n with property P , then there are at most ( n !) f P ( n ) < 2 f P ( n ) n log n subgroups with P . Colva Roney-Dougal University of St Andrews Generation of finite groups

  69. More on random subgroups of S n P – property of permutation groups. If have a bound f P ( n ) on the number of generators of a subgroup of S n with property P , then there are at most ( n !) f P ( n ) < 2 f P ( n ) n log n subgroups with P . Corollary n P – property such that f P ( n ) < (log n ) 1+ ε for ε > 0 . Colva Roney-Dougal University of St Andrews Generation of finite groups

  70. More on random subgroups of S n P – property of permutation groups. If have a bound f P ( n ) on the number of generators of a subgroup of S n with property P , then there are at most ( n !) f P ( n ) < 2 f P ( n ) n log n subgroups with P . Corollary n P – property such that f P ( n ) < (log n ) 1+ ε for ε > 0 . Then the proportion of subgroups of S n that satisfy P Colva Roney-Dougal University of St Andrews Generation of finite groups

  71. More on random subgroups of S n P – property of permutation groups. If have a bound f P ( n ) on the number of generators of a subgroup of S n with property P , then there are at most ( n !) f P ( n ) < 2 f P ( n ) n log n subgroups with P . Corollary n P – property such that f P ( n ) < (log n ) 1+ ε for ε > 0 . Then the proportion of subgroups of S n that satisfy P tends to 0 as n → ∞ . Colva Roney-Dougal University of St Andrews Generation of finite groups

  72. More on random subgroups of S n P – property of permutation groups. If have a bound f P ( n ) on the number of generators of a subgroup of S n with property P , then there are at most ( n !) f P ( n ) < 2 f P ( n ) n log n subgroups with P . Corollary n P – property such that f P ( n ) < (log n ) 1+ ε for ε > 0 . Then the proportion of subgroups of S n that satisfy P tends to 0 as n → ∞ . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant b such that the number of transitive subgroups of S n Colva Roney-Dougal University of St Andrews Generation of finite groups

  73. More on random subgroups of S n P – property of permutation groups. If have a bound f P ( n ) on the number of generators of a subgroup of S n with property P , then there are at most ( n !) f P ( n ) < 2 f P ( n ) n log n subgroups with P . Corollary n P – property such that f P ( n ) < (log n ) 1+ ε for ε > 0 . Then the proportion of subgroups of S n that satisfy P tends to 0 as n → ∞ . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant b such that the number of transitive subgroups of S n is at most 2 bn 2 / √ log n . Colva Roney-Dougal University of St Andrews Generation of finite groups

  74. More on random subgroups of S n P – property of permutation groups. If have a bound f P ( n ) on the number of generators of a subgroup of S n with property P , then there are at most ( n !) f P ( n ) < 2 f P ( n ) n log n subgroups with P . Corollary n P – property such that f P ( n ) < (log n ) 1+ ε for ε > 0 . Then the proportion of subgroups of S n that satisfy P tends to 0 as n → ∞ . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant b such that the number of transitive subgroups of S n is at most 2 bn 2 / √ log n . Hence the proportion of subgroups of S n that are transitive tends to 0 as n → ∞ . Colva Roney-Dougal University of St Andrews Generation of finite groups

  75. More on random subgroups of S n P – property of permutation groups. If have a bound f P ( n ) on the number of generators of a subgroup of S n with property P , then there are at most ( n !) f P ( n ) < 2 f P ( n ) n log n subgroups with P . Corollary n P – property such that f P ( n ) < (log n ) 1+ ε for ε > 0 . Then the proportion of subgroups of S n that satisfy P tends to 0 as n → ∞ . Theorem (Lucchini, Menegazzo, Morigi 00) There exists a constant b such that the number of transitive subgroups of S n is at most 2 bn 2 / √ log n . Hence the proportion of subgroups of S n that are transitive tends to 0 as n → ∞ . Colva Roney-Dougal University of St Andrews Generation of finite groups

  76. Some speculation Colva Roney-Dougal University of St Andrews Generation of finite groups

  77. Some speculation It looks likely that a random subgroup of S n should be Colva Roney-Dougal University of St Andrews Generation of finite groups

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