ALTERNATING GROUP Meaning and
Definition
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The alternating group, denoted by A_n, is a mathematical concept in group theory that represents the set of even permutations of a finite set containing n elements. An even permutation is one that can be achieved by an even number of swaps or transpositions of elements.
To understand the alternating group, it is necessary to define some key components. Firstly, a permutation is a bijective mapping of the set's elements onto itself. It can be visualized as rearranging the elements in a particular order. In group theory, permutations are considered as elements of a group.
The alternating group specifically focuses on even permutations. An even permutation can be decomposed into an even number of transpositions, i.e., swapping of two elements. This means that any permutation forming the alternating group can be written as a product of an even number of transpositions.
The alternating group has numerous interesting properties and applications. It plays a crucial role in understanding symmetry and permutations. For instance, it is used in combinatorics, abstract algebra, and the study of Rubik's cube.
In summary, the alternating group A_n refers to the set of even permutations of a finite set with n elements. It represents a subset of permutations that can be expressed as a product of an even number of transpositions, highlighting their importance in the realm of symmetry and mathematical structures.
Common Misspellings for ALTERNATING GROUP
- zlternating group
- slternating group
- wlternating group
- qlternating group
- akternating group
- apternating group
- aoternating group
- alrernating group
- alfernating group
- algernating group
- alyernating group
- al6ernating group
- al5ernating group
- altwrnating group
- altsrnating group
- altdrnating group
- altrrnating group
- alt4rnating group
- alt3rnating group
- alteenating group
Etymology of ALTERNATING GROUP
The word "alternating" in "alternating group" comes from the mathematical concept of permutation, which is a rearrangement of elements. In combinatorics, an alternating permutation is one that can be decomposed into an even number of adjacent transpositions, or swaps of adjacent elements.
The term "alternating group" originated from the study of symmetries and permutations, particularly in the context of group theory. The alternating group is a specific type of permutation group, denoted as "A_n", that consists of even permutations. An even permutation is one that can be expressed as a product of an even number of transpositions.
The word "alternating" is used to denote this subgroup of even permutations, as it reflects the property that these permutations can be broken down into an alternating sequence of swaps.
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