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The identity permutation is an even permutation. [1] An even permutation can be obtained as the composition of an even number (and only an even number) of exchanges (called transpositions) of two elements, while an odd permutation can be obtained by (only) an odd number of transpositions.
A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]
The identity permutation is defined by () = for all elements , and can be denoted by the number , [a] by =, or by a single 1-cycle (x). [ 15 ] [ 16 ] The set of all permutations of a set with n elements forms the symmetric group S n {\displaystyle S_{n}} , where the group operation is composition of functions .
Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ , or less commonly the Latin lower case e .
When π is the identity permutation, which has () = for all i, both C π and R π are the identity matrix. There are n ! permutation matrices, since there are n ! permutations and the map C : π ↦ C π {\displaystyle C\colon \pi \mapsto C_{\pi }} is a one-to-one correspondence between permutations and permutation matrices.
The specific nature of the outer automorphism is as follows. The 360 permutations in the even subgroup (A 6) are transformed amongst themselves: the sole identity permutation maps to itself; a 3-cycle such as (1 2 3) maps to the product of two 3-cycles such as (1 4 5)(2 6 3) and vice versa, accounting for 40 permutations each way;
The identity is its minimum, and the permutation formed by reversing the identity is its maximum. If a permutation were assigned to each inversion set using the element-based definition, the resulting order of permutations would be that of a Cayley graph, where an edge corresponds to the swapping of two elements on consecutive places. This ...
This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a cyclic group and is thus abelian. In Galois theory, this corresponds to the fact that the quadratic formula gives a direct solution to the general quadratic polynomial after extracting only a single root.