Parker vector

 ( Permutation Groups ) 

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

---

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:

$P\_k\; =\; \backslash frac\{k\}$ \sum_{g \in G} c_k(g) where c _k( g) is the number of k-cycles in the cycle decomposition of g.

---

Examples For the group of even permutations on three elements, the Parker vector is (1,0,2). The group of all permutations on three elements has Parker vector (1,1,1). For any of the subgroups of the above with just two elements, the Parker vector is (2,1,0).The trivial subgroup has Parker vector (3,0,0).

---

Applications

• Peter J. Cameron (1999). 9780521653787, Cambridge University Press. . ISBN 9780521653787
• Aart Blokhuis (2026). 9780792369943, Springer. . ISBN 9780792369943

Post Comment