Parker vector explained

In mathematics, especially the field of group theory, the Parker vector is an integer vector that describes a permutation group in terms of the cycle structure of its elements.

Definition

The Parker vector P of a permutation group G acting on a set of size n, is the vector whose kth component for k = 1, ..., n is given by:

P _{k} = (k)/(|G|) \sum _{g} c _{k(g)} where c_k(g) is the number of k-cycles in the cycle decomposition of g.

Applications

The Parker vector can assist in the recognition of Galois groups.

References

• Book: Permutation Groups . Peter J. Cameron . Cambridge University Press . 1999 . 0-521-65378-9 . registration . 48 . Parker Vector. .
• Book: Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference . Aart Blokhuis . Springer . 2001 . 0-7923-6994-7 .