EVEN PERMUTATION Meaning and
Definition
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An even permutation is a mathematical concept used in the field of group theory, specifically in the study of permutations. A permutation is a rearrangement of a set of objects, typically represented by a sequence of numbers or symbols. In the context of even permutations, these permutations are classified according to a certain property.
In a permutation, if the total number of swaps required to bring the elements back to their original order is an even number, then it is considered an even permutation. Alternatively, an even permutation can also be defined as a permutation that can be decomposed into an even number of transpositions, where a transposition is an interchanging of two elements.
For example, let's consider the permutation (1 4 2 3). We observe that it can be decomposed into two transpositions: (1 4) and (4 2). Since there are two transpositions, which is an even number, this permutation is classified as an even permutation.
Even permutations have interesting properties and play a fundamental role in various mathematical fields, such as abstract algebra and geometry. They form a subgroup of the symmetric group, a group consisting of all possible permutations of a given set. Understanding even permutations is crucial in studying more complex mathematical structures and concepts related to symmetry, such as alternating groups, group actions, and matrix representations.
Common Misspellings for EVEN PERMUTATION
- wven permutation
- sven permutation
- dven permutation
- rven permutation
- 4ven permutation
- 3ven permutation
- ecen permutation
- eben permutation
- egen permutation
- efen permutation
- evwn permutation
- evsn permutation
- evdn permutation
- evrn permutation
- ev4n permutation
- ev3n permutation
- eveb permutation
- evem permutation
- evej permutation
- eveh permutation
Etymology of EVEN PERMUTATION
The word "even permutation" is derived from the term "permutation", which comes from the Latin word "permutatio", meaning "act of changing or exchanging". In mathematics, a permutation refers to the rearrangement or ordering of a set of objects.
The term "even" in "even permutation" is used to describe a specific property of permutations. In the context of permutations, an even permutation refers to a permutation that can be decomposed into an even number of transpositions. A transposition is a permutation that swaps two elements.
The concept of even and odd permutations emerged from the study of the symmetric group, which consists of all possible permutations of a finite set. By analyzing the structure of permutations, mathematicians began categorizing them based on their properties, and the terms "even" and "odd" were introduced to distinguish between two distinct types of permutations.
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