Groups {substitution group} {permutation group} can have operation that maps one element to another element {substitution, group}. An example is rearranging element positions {permutation, group}, such as exchanging elements at line-segment ends. Permutations can result in equivalent subsets {combination, group}. Product element is in group {closure, group} {closed group}.
identity
Substitution groups must have identity operation.
inverse
Substitution-group operations must have inverse operations.
transitive
Group operation, such as rotations, can substitute all elements {transitive group}. Group operations, such as division, may not apply to all elements or some elements may not be operation results {intransitive group}.
primitive
Elements can have separate subsets that permute among themselves {imprimitive group} or have no separable subsets {primitive group}.
abstract groups
Generalized substitution groups {abstract group} show group properties.
Mathematical Sciences>Group Theory>Mathematical Groups
3-Group Theory-Mathematical Groups
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0224