# Alternating Group: Show στσ-1τ-1 ∈ An

• PhysicsUnderg
In summary, the alternating group, denoted by An, is a subgroup of the symmetric group Sn consisting of all even permutations. A permutation is a rearrangement of a set of elements, and 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, and to show that it is in An, it must be proven to be 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 related to the alternating group An as it is an element of the group and is used in proving various
PhysicsUnderg
For any elements σ, τ ∈ Sn, show that στσ-1τ-1 ∈ An.

What did you try already?? Just calculate the sign of the element.

## 1. What is the alternating group?

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.

## 2. What is a permutation?

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.

## 3. What is στσ-1τ-1?

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-1τ-1.

## 4. How do you show that στσ-1τ-1 is an element of An?

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.

## 5. How is στσ-1τ-1 related to the alternating group?

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.

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