# Questions tagged [permutation-cycles]

For elementary questions concerning permutation cycles and permutation groups. This includes all representations of permutations (two-line arrays, cycles, bipartite graphs); transpositions and the sign/parity of a permutation; the Symmetric group and the Alternating group. To be used with the (permutations) tag to make the distinction between abstract algebra permutation questions and combinatoric permutation questions.

122 questions
1answer
18 views

### permutations cycle

I am doing abstract algebra problems, but unfortunately, the book we are using for the class is quite poor and leaves out lots of definitions and explanations, so I am not even sure if I completely ...
1answer
11 views

### Number of times we have to compose a permutation in order to have exactly k fixed points

Let $f = (1 4 6)(2 7 5 8 10)(3 9)$ in $S_{10}$. Find an integer $n$ such that $f^n$ has exactly $7$ fixed points. I provided the exact numbers, but would welcome a more general solution.
1answer
38 views

2answers
85 views

### Count the number of permutations of certain cycles type

Suppose we have $n$ elements, assume there is a permutations over $k$ elements among the $n$ elements so $n-k$ are fixed. Let that the permutation over the k elements is represented by permutation ...
1answer
22 views

### Permutations in $S_9$ with length $4,3,2$.

How many permutations in $S_9$ have one cycle of length $4$, one of length $3$, and one of length $2$? My attempt so far is: ${9 \choose 4}*{5 \choose 3}*{2 \choose 2}$, which is the way to choose ...
1answer
18 views

### Confused about composition of two permutation cycles

I'm slightly confused about the product of the following permutation cycles. I am given that $s_1 = (1\ 2)$ and $s_2 = (2\ 3)$ where both are generators for the symmetric group $S_3$. My textbook ...
1answer
38 views

### The permutation is given. For how many functions $f:\Bbb{N_{10}} \rightarrow \Bbb{N_{10}}$ are $f(\pi(i))=\pi(f(i))$?

The permutation $\pi\in S_{10}$ is given by the table: \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline i & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \pi &...
0answers
30 views

2answers
38 views

### Compositions & Transpositions of permutations

Consider the set of all permutations $S_n$. Fix an element $\tau\in S_n$. Then the sets $\{\sigma\circ\tau\mid \sigma\in S_n\}= \{\tau \circ\sigma\mid \sigma\in S_n\}$ have exactly $n!$ elements. ...
1answer
28 views

### Given a cycle $c \in S_n$ with $ord(c) = s$ and $s = kt$, prove that $c^k$ is a product of $k$ cycles of length $t$. [closed]

I came across this question in a recent exam. Given that $ord(c) = s$, we assume that $c^s = c^{kt} = (id) \implies (c^{k})^t = (id)$. That means that $c^k$ is a cycle of order $t$. Can you ...
1answer
34 views

### On terminology; what is meaning of the “decrement” of a permuation? (or what is the alternative word or phrase used for this definition)

I am currently studying some lecture notes in Russian and I frequently look up the theorems and topics in English for better understanding (I'm not a native Russian so I merely just use the Russian ...
2answers
74 views

### In how many ways can 3 employees visit 40 locations

Three employees need to visit 40 different cities under the following conditions: each location should be visited by exactly one employee, and no location should be visited multiple times. The travel ...
1answer
29 views

### Permutation as string position recording in Wilson's FSG book

I'm reading Wilson's The Finite Simple Group book, mainly the Alternating group chapter, which explains the splitting criterion quite in detail. While heading there, I came across a possible ...
1answer
49 views

### On the product of two cycles (and its conjugates)

So this is from Charles C. Pinter's "A Book of Abstract Algebra"- specifically, it's from the second chapter on permutations. The question is: Let $\alpha_1$ and $\alpha_2$ be cycles of the same ...
1answer
46 views

### Geometrical meaning of A4 conjugacy classes of elements of order 3

I know that elements of order 3 in A4 group split into two separate conjugacy classes, due to lack of odd elements to form a unique class like happens in the full S4. Since A4 is the rotation ...