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.
Read more about this topic: Generic Polynomial
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