Parker vector

This article is about a mathematical vector. For the pen, see Parker Pen Company § 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:

$P_{k}={\frac {k}{|G|}}\sum _{g\in G}c_{k}(g)$ 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

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