Permutation Expressions: Understanding and Computing

In summary, permutations in abstract algebra are rearrangements of a set of elements represented by a function. They are typically written using cycle notation and the order of a permutation is the number of elements being rearranged. Permutations are used in abstract algebra to study the structure and properties of groups, and there are special types such as identity, inverse, even, and odd permutations.
  • #1
kathrynag
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Homework Statement



Compute the expression shown for the permutations
1.[tex]\left|<\phi>\right|[/tex]
2..[tex]\left|<\tau^2>\right|[/tex]
3.[tex]\phi^{100}[/tex]
where:
[tex]\phi[/tex]= top row:1, 2 , 3 ,4 , 5 ,6
bottom row: 3,1, 4,5,6,2

[tex]\tau[/tex] = top row: 1,2,3,4,5,6
bottom row: 2,4,1,3,6,5

Homework Equations





The Attempt at a Solution


Ok, my main problem is that I don't even know what they're asking. I understand how to do permutation multiplication and composition, but not this.
I do know from the back of the book that #2 is 2, but I want to know why this is true.
 
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  • #2
Does it just mean finding the identity?
 

What are permutations in abstract algebra?

In abstract algebra, permutations are defined as a rearrangement of a set of elements, where each element appears only once and every element in the set is included in the rearrangement. This rearrangement is represented by a function, also known as a permutation, that maps the original set to the rearranged set.

How are permutations written in abstract algebra?

In abstract algebra, permutations are typically written using cycle notation, where the elements being rearranged are represented by numbers or symbols within parentheses. For example, the permutation (1 2 3) would represent the rearrangement of the elements 1, 2, and 3.

What is the order of a permutation in abstract algebra?

The order of a permutation in abstract algebra is the number of elements being rearranged. For example, the permutation (1 2 3) has an order of 3, as it rearranges 3 elements.

How are permutations used in abstract algebra?

Permutations are used in abstract algebra to study the structure and properties of groups, which are mathematical structures consisting of a set of elements and a binary operation that satisfies certain properties. Permutations are also used in other branches of mathematics, such as combinatorics and graph theory.

Are there any special types of permutations in abstract algebra?

Yes, there are several special types of permutations in abstract algebra, including identity permutations, which leave all elements unchanged, and inverse permutations, which undo the rearrangement of another permutation. Other types include even and odd permutations, which are based on the number of transpositions (swaps) needed to create the permutation.

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