Parker vector

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:

[math]\displaystyle{ P_k = \frac{k}{|G|} \sum_{g \in G} c_k(g) }[/math] 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

• Peter J. Cameron (1999). Permutation Groups. Cambridge University Press. p. 48. ISBN 0-521-65378-9. https://archive.org/details/permutationgroup0000came. "Parker Vector." 
• Aart Blokhuis (2001). Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference. Springer. ISBN 0-7923-6994-7. https://books.google.com/books?id=P8Ra5zHVzQUC&dq=%22Parker+Vector%22&pg=PA59.