- #1
PhysicsUnderg
- 19
- 0
For any elements σ, τ ∈ S_{n}, show that στσ^{-1}τ^{-1} ∈ A_{n}.
The alternating group, denoted by An, is a subgroup of the symmetric group Sn consisting of all even permutations. In other words, it contains all permutations that can be written as a product of an even number of transpositions.
A permutation is a rearrangement of a set of elements. In the alternating group, the elements being rearranged are the integers from 1 to n.
στσ-1τ-1 is a conjugate of the element τ by the element σ, where both σ and τ are permutations in An. This can be represented as στσ-1τ-1 = στσ^{-1}τ^{-1}.
To show that στσ-1τ-1 is in An, we need to prove that it is an even permutation. This can be done by showing that it can be written as a product of an even number of transpositions.
στσ-1τ-1 is an element of the alternating group An, meaning it follows the same properties and rules as other elements in the group. It is also used in proving various theorems and properties of the alternating group.