There are functions μ1, μ2: ℕ → ℕ, such that any group, all of whose subgroups, which can be generated by μ2(n) elements, are n-step subnormal subgroups, is nilpotent of class ≤ μ1(n).[1] It is defined by:

  • μ₁(1) = 2
  • μ₁(n) = μ₆(μ₇(n)) (n>1)
  • μ₂(1) = 1
  • μ₂(n) = μ₂(n-1) + μ₇(n) (n>1)
  • μ₄(n,1) = 1
  • μ₄(n,2) = 1 + 2ⁿ⁻¹ + μ₉(n, μ₈(n))
  • μ₄(n,m) = μ₁₂(μ₄(n,m-1), μ₄(n,2)) (m>2)
  • μ₅(n) = n μ₁₃(n)
  • μ₆(n) = μ₄(n, μ₅(n))
  • μ₇(n) = 1 + μ₄(n, (n-1) μ₁(n-1)) (n>1)
  • μ₈(1) = 1
  • μ₈(n) = n (μ₈(n-1))² (n>1)
  • μ₉(1,m) = 2
  • μ₉(n,m) = n - 1 + μ₁₁(n,m) (n>1)
  • μ₁₀(n) = n + n² + … + n
  • μ₁₁(n,m) = ⌊μ₁₀(n) log₂(m)⌋ + 1
  • μ₁₂(c,d) = (c+1)cd/2 - c(c-1)/2
  • μ₁₃(n) = n + μ₁₅(n)
  • μ₁₄(n) = n + n² + … + nⁿ
  • μ₁₅(n) = μ₁₆(μ₁₄(n)) + 1
  • μ₁₆(n) = ½n(5n-1) + 1

Sources

  1. Roseblade, J. E.. On Groups in which Every Subgroup is Subnormal. Retrieved 2016-10-17.
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