Generic Polynomial - Groups With Generic Polynomials

Groups With Generic Polynomials

  • The symmetric group Sn. This is trivial, as

is a generic polynomial for Sn.

  • Cyclic groups Cn, where n is not divisible by eight. Lenstra showed that a cyclic group does not have a generic polynomial if n is divisible by eight, and Smith explicitly constructs such a polynomial in case n is not divisible by eight.
  • The cyclic group construction leads to other classes of generic polynomials; in particular the dihedral group Dn has a generic polynomial if and only if n is not divisible by eight.
  • The quaternion group Q8.
  • Heisenberg groups for any odd prime p.
  • The alternating group A4.
  • The alternating group A5.
  • Reflection groups defined over Q, including in particular groups of the root systems for E6, E7, and E8
  • Any group which is a direct product of two groups both of which have generic polynomials.
  • Any group which is a wreath product of two groups both of which have generic polynomials.

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