Linked Questions

2
votes
1answer
245 views

How do you interpret a product of transpositions? [duplicate]

I'm trying to understand why the product of transpositions for a specific permutation is not unique. Intuitively, it somewhat makes sense to me since I can get the answer but I don't actually know why ...
0
votes
1answer
57 views

Write a permutation as a product of transpositions. [duplicate]

Let $\alpha = (1 \ 6 \ 3) (2 \ 9) (4 \ 8 \ 10) \in S_{10}$ be a permutation. Write $\alpha$ as a product of transpositions, i.e. of cyclic permutations of order 2. Note that transpositions do not need ...
0
votes
0answers
19 views

Why does $A_6$ 6-cycle from $S_6$ which also has even number of 2-cycles and hence is even permutation [duplicate]

For $S_6$,the possible orders are 1,2,3,4,5,6. While for $A_6$, the possible orders are 1,2,3,4,5 but not 6. Why does $A_6$ neglects the 6-cycle in $S_n$ which also has 0 number of 2-cycles and ...
7
votes
3answers
1k views

Product of permutation cycles, transpositions. Are there different conventions in the order?

From this answer I get that within each cycle you map each element to the one on the right, when taking the product of cycles the one on the right should be performed first, as a typical operator. ...
2
votes
2answers
343 views

Permutation as a Product of $2$ cycles

\begin{bmatrix}1&2&3&4&5&6&7&8\\2&3&4&5&1&7&8&6\end{bmatrix} I have already written this permutation as disjoint cycles: (12345)(678) My ...
1
vote
1answer
902 views

Why is this the method to getting transpositions from disjoint cycles?

I have the disjoint cycle: $$(156)(2437).$$ Apparently the "method" would get us: $$(1,6)(1,5)(2,7)(2,3)(2,4).$$ Basically you take the first number, and put it as a transposition of the last number ...
-2
votes
2answers
468 views

To find order of permutation

Let $\sigma$ be the permutation given by Is their a short way to do this.Thanks
0
votes
1answer
305 views

Decomposing a permutation into multiplication of transpositions [duplicate]

I have a permutation in cyclic notation, for example $(132)$, and i want to represent it as multiplication of transpositions. What is the fastest way to do it?
0
votes
2answers
162 views

Every permutation of $n$ elements is a product of transpositions of the $n$ elements.

Every permutation of $n$ elements is a product of transpositions of the $n$ elements. My work: We proceed by induction on $n$. Ovbiusly this stetement is true if $n=1,2$. Now, suppose that $n\geq 3$ ...
1
vote
2answers
122 views

Classifying permutations in terms of their cycle notation

Is there are a standard way of referring to permutations in terms of their cycle notation? For example: Does the set of all permutations in $S_4$ that can be expressed as the composition of two two-...
0
votes
0answers
211 views

Finding the Order of a Permutation

I feel so lost on finding the order of a permutation. I understand the definition is the # of transpositions it can be "broken down" into, but how do I go about actually finding these transpositions? ...
0
votes
2answers
92 views

Permutation written as a single cycle

How do I rewrite $(1\,2)(1\,3)(1\,4)(1\,5)$ as a single cycle? I have tried questions in the form: $(1\,4\,3\,5\,2)(4\,5\,3\,2\,1)$.