Symmetric and Alternating Groups disjoint cycles

In summary, the given proof shows that for disjoint cycles a and b in Sn, ab = ba. This is because for any element i in T, both a and b map i to itself, making the compositions of a and b in either order equal. Therefore, ab = ba holds true for disjoint cycles in Sn.
  • #1
tehme2
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Homework Statement



Let a = (a1a2..ak) and b = (c1c2..ck) be disjoint cycles in Sn. Prove that ab = ba.

The Attempt at a Solution


Sn consists of the permutations of the elements of T where T = {1,2,3,...,n}
so assume we take an i from T. Then either i is in a, i is in b, or i is in neither a or b

1. assume i is in a.
then a o b(i) = a(i) as b(i) = i since b maps it to itself
then b o a(i) = b(a(i)) = a(i) as b maps this a(i) to itself

2. assume i is in b.

then a o b (i) = a(b(i)) = b(i) as a maps this b(i) to itself
then b o a(i) = b(i)

3. assume it is in neither

then a o b (i) = a(i) = i as both a and b map i to itself
then b o a (i) = b(i) = i

so regardless of which permutation i is in, we see ab = ba for disjoint cycles a and b.
 
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  • #2
Your proof is fine. Is there a question about this?
 

FAQ: Symmetric and Alternating Groups disjoint cycles

1. What is a symmetric group?

A symmetric group is a group of permutations of a set, where the order of elements in the set does not matter. In other words, the group contains all possible ways to rearrange the elements of the set without changing its overall structure.

2. What is an alternating group?

An alternating group is a subgroup of a symmetric group, where all the even permutations are included. In other words, it contains all the even permutations of a set, while omitting the odd permutations.

3. How are symmetric and alternating groups related?

Symmetric and alternating groups are closely related as alternating groups are a special case of symmetric groups. Every alternating group is a subgroup of a symmetric group, but not every symmetric group is an alternating group.

4. What are disjoint cycles in symmetric and alternating groups?

Disjoint cycles in symmetric and alternating groups refer to permutations that do not overlap or share any elements. For example, the permutation (1 2 3)(4 5) is a disjoint cycle since the first cycle (1 2 3) only affects elements 1, 2, and 3, and the second cycle (4 5) only affects elements 4 and 5.

5. How can disjoint cycles be used to represent elements in symmetric and alternating groups?

Disjoint cycles are a useful way to represent elements in symmetric and alternating groups because they provide a concise and unique representation of a permutation. They are also useful in performing operations on permutations, such as finding the inverse of a permutation or composing two permutations together.

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