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 "k"th component for "k"=1…"n" is given by::$$P_k = frac{k} sum_{g in G} c_k(g) where "c"_"k"("g") is the number of "k"-cycles in the cycle decomposition of "g".

Examples

Applications

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

References

*cite book | title = Permutation Groups | author = Peter J. Cameron | publisher = Cambridge University Press | year = 1999 | isbn = 0521653789 | url = http://books.google.com/books?id=4bNj8K1omGAC&pg=PA48&dq=%22Parker+Vector%22&lr=&as_brr=0&ei=JeKoR8iKHIvsiQHfjNWqCg&sig=v2x98oF0YEsxyNm8HXq0BHPDXPA
*cite book | title = Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference | author = Aart Blokhuis | publisher = Springer | year = 2001 | isbn = 0792369947 | url = http://books.google.com/books?id=P8Ra5zHVzQUC&pg=PA59&dq=%22Parker+Vector%22&lr=&as_brr=0&ei=JeKoR8iKHIvsiQHfjNWqCg&sig=yjCRM55fuSmS84OZ-ugc9tT2kyo