Parker vector
Encyclopedia
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.
where ck(g) is the number of k-cycles in the cycle decomposition of g.
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially the field of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the Parker vector is an integer vector that describes a permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
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: where ck(g) is the number of k-cycles in the cycle decomposition of g.Applications
The Parker vector can assist in the recognition of Galois groupGalois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
s.