PERMUTATION GROUP Meaning and
Definition
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A permutation group is a mathematical concept that represents a collection or set of all the possible permutations of a given set. In other words, it is a group of all the ways in which the elements of a set can be rearranged or permuted.
Formally, a permutation group is a set of bijective mappings, also known as permutations, on a particular set. These permutations preserve the structure of the set, meaning they only change the ordering of its elements. The elements of the group can be obtained by composing or multiplying these permutations together.
A permutation group is often denoted using cycle notation, where each permutation is expressed as a product of disjoint cycles. These cycles represent the movement or swapping of elements within the set, exhibiting the transformation that occurs when the permutation is applied.
Permutation groups have various properties and features that make them significant in diverse areas of mathematics, such as algebra, combinatorics, and graph theory. They are extensively studied in the field of group theory, where their properties and symmetries are thoroughly examined.
Permutation groups provide a powerful framework for analyzing and understanding symmetric structures, patterns, and arrangements. They are widely used in various applications, including cryptography, coding theory, network optimization, statistical analysis, and theoretical computer science. Ultimately, permutation groups play a fundamental role in investigating the symmetries and transformations of objects and systems across numerous mathematical and scientific domains.
Common Misspellings for PERMUTATION GROUP
- oermutation group
- lermutation group
- 0ermutation group
- pwrmutation group
- psrmutation group
- pdrmutation group
- prrmutation group
- p4rmutation group
- p3rmutation group
- peemutation group
- pedmutation group
- pefmutation group
- petmutation group
- pe5mutation group
- pe4mutation group
- pernutation group
- perkutation group
- perjutation group
- permytation group
- permhtation group
Etymology of PERMUTATION GROUP
The word "permutation" comes from the Latin word "permutatio", which means "act of changing". It is derived from the roots "per", meaning "through" or "thoroughly", and "mutare", meaning "to change".
The word "group" comes from the Old English word "gryppe", which means "band" or "troop". It is related to the Old Norse word "grup", meaning "to catch" or "to grasp".
The term "permutation group" combines these two words to describe a mathematical concept that deals with the changing or rearranging of elements within a set, and is represented by a group structure.
