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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
468 views

### To find order of permutation

Let $\sigma$ be the permutation given by Is their a short way to do this.Thanks
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?
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 $n\geq 3$ ...
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-...
How do I rewrite $(1\,2)(1\,3)(1\,4)(1\,5)$ as a single cycle? I have tried questions in the form: $(1\,4\,3\,5\,2)(4\,5\,3\,2\,1)$.