# Conjugation of a permutation by a permutation in a permutation group

• karnten07
In summary, the theorem states that in a permutation group, the inverse permutation is the result of conjugation by another permutation. This means that if we have a cycle <a1,...,as> and an arbitrary permutation, their conjugation will result in a new cycle with elements being the images of the original elements under the permutation. A proof for this can be found by trying out specific permutations and observing the pattern.
karnten07

## Homework Statement

Let n $$\geq$$1. Let <a1,...,as> $$\in$$Sn be a cycle and let $$\sigma$$$$\in$$Sn be arbitrary. Show that

$$\sigma\circ$$ <a1,...,as> $$\circ$$$$\sigma^{-1}$$ = <$$\sigma$$(a1),...,$$\sigma$$(as)> in Sn.

## The Attempt at a Solution

As the title says, i believe this is a theorem regarding that the inverse permutation is the effect of a conjugation of a permutation by a permutation in a permutation group. Does anyone know a proof for this or where to find one?

That's pretty hard to read. It's usually better if you put adjacent things in one [tex] tag instead of several.

morphism said:
That's pretty hard to read. It's usually better if you put adjacent things in one [tex] tag instead of several.

Im sorry, here it is written in wikipedia simpler:

One theorem regarding the inverse permutation is the effect of a conjugation of a permutation by a permutation in a permutation group. If we have a permutation Q=(i1 i2 … in) and a permutation P, then PQP−1 = (P(i1) P(i2) … P(in)).

Try it out with specific permutations and see if you can spot a general pattern.

## Q1. What is conjugation of a permutation?

Conjugation of a permutation is a process in which a permutation is transformed by another permutation. This is done by applying the second permutation to the elements of the first permutation, resulting in a new permutation.

## Q2. What is a permutation group?

A permutation group is a group of permutations that can be combined together through operations such as composition and inversion. These groups are commonly used in mathematics, particularly in the study of symmetry and group theory.

## Q3. How does conjugation work in a permutation group?

In a permutation group, conjugation is performed by applying a permutation to another permutation, resulting in a new permutation that has the same structure as the original one. This new permutation is said to be conjugate to the original permutation.

## Q4. What is the purpose of conjugation in a permutation group?

The purpose of conjugation in a permutation group is to transform a permutation into another permutation with the same structure, but possibly with different elements in different positions. This allows for easier analysis and simplification of permutations within a group.

## Q5. Can any two permutations be conjugate to each other in a permutation group?

No, not all permutations can be conjugate to each other in a permutation group. The two permutations must have the same structure, meaning they must have the same number of elements and cycles. Additionally, the two permutations must be in the same group and have a common element that can be used for conjugation.

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