Show if Odd n-cycles Equal Conjugacy Classes A_n

Thread starter abm
Start date Oct 31, 2005
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In summary, an n-cycle is a permutation of n elements in a specific order, and in this concept, it refers to a cycle of n elements in a symmetric group. Conjugacy in a group is the relationship between two elements where one can be transformed into the other by a group operation. Odd n-cycles in a symmetric group are equal in conjugacy classes, meaning they can all be transformed into each other. This concept relates to the alternating group A_n, which is a subgroup of the symmetric group. For example, in the symmetric group S_4, the two conjugacy classes of odd 3-cycles, (123) and (132), are conjugate to each other, but not to any even permutations in the

Oct 31, 2005

abm

Could anyone help to show that if n is odd then the set of all n-cycles consists of two conjugacy classes of equal size in A_n? Thx

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Nov 5, 2005

quantumdude

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Show us how you started and where you got stuck.

FAQ: Show if Odd n-cycles Equal Conjugacy Classes A_n

1. What is an "n-cycle" in the context of this concept?

An n-cycle is a permutation of n elements in a specific order. In the context of this concept, it refers to a cycle of n elements in a symmetric group, where each element is mapped to the next element in the cycle.

2. How is conjugacy defined in a group?

In group theory, conjugacy refers to the relationship between two elements in a group, where one element can be transformed into the other by a group operation. For example, in a symmetric group, two n-cycles are conjugate if they have the same cycle structure.

3. What does it mean for odd n-cycles to be equal in conjugacy classes?

This means that all odd n-cycles in a symmetric group are conjugate to each other. In other words, they can all be transformed into one another by a group operation.

4. How does this concept relate to the alternating group A_n?

The alternating group A_n is a subgroup of the symmetric group, consisting of even permutations. In this concept, we are specifically looking at odd n-cycles in the symmetric group, which are not in the alternating group. However, the conjugacy classes of odd n-cycles are the same in both groups.

5. Can you provide an example to illustrate this concept?

Sure, let's consider the symmetric group S_4. There are two conjugacy classes of odd 3-cycles: (123) and (132). These two cycles are conjugate to each other, meaning they can be transformed into one another by a group operation. However, they are not conjugate to any even permutations, which are in the alternating group A_4.