The basic concept of permutation group;
A permutation group is a group generated by permutation, in which every element is a permutation. Replacement is the process of rearranging elements in a collection, which can be regarded as the mapping of elements in the collection. Permutations in permutation groups can be different or the same, but they must satisfy certain associative laws.
Properties of permutation groups:
Permutation groups have some important properties. Permutation groups are groups, so they have the basic properties of groups, such as closure, associative law, unit element, inverse element and so on. Permutations in permutation groups can be regarded as reversible operations, so permutation groups are also reversible. Permutation groups are also transitive, that is, in a permutation group, if the permutation times of two elements are equal.
Classification of permutation groups:
According to the different properties of permutation in permutation groups, permutation groups can be divided into different types. One of the important permutation groups is cyclic permutation group, which is a group generated by a permutation cycle. A cyclic permutation group has only finite elements, so it is a finite group. Another important permutation group is Abel permutation group, which is generated by Abel permutation. Abel permutation refers to those permutations that can be transformed into each other through a series of reflection transformations.
Representation of permutation groups;
In practical application, it is usually necessary to use mathematical symbols or graphics to represent permutation groups. The common representation method is matrix representation, that is, each arrangement is regarded as a matrix, and then the combination of arrangements is calculated by matrix multiplication. Another common representation is word representation, that is, each arrangement is regarded as a word, and then they are connected to form a long string.
Calculation method of permutation group;
In the calculation of permutation groups, some basic calculation methods are usually needed. For example, the properties of permutation groups can be understood by calculating the order of each element, the structure of permutation groups can be understood by calculating the yoke class, and the centralized substructure of permutation groups can be studied by calculating the centralized substructure.