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Mathematics Stack Exchange

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.

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 to make the distinction between abstract algebra permutation questions and combinatoric permutation questions.

666 questions
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What's $(1 2 3)(1 4 5)$? Everybody gives a different answer.

From my calculation: $(1 4 5 2 3)$. From Joseph Gallian's Contemporary Modern Algebra, 9th edition, page 100: From WolframAlpha:
daedsidog
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How is the permutation relationship interpreted

I was given the example as an illustration of structure of permutations in my lecture notes on algebra as shown below: $\bigl(\begin{smallmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 2 & 1 & 3 & 4 & 9 & 6 & 5 & 8 & 7 \end{smallmatrix}\bigr)$ $=…
Gvxfjørt
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Circular Permutation Problem with overlapping patterns

Intro: We know that if there are two symbols, e.g. a and b, giving two spaces to place these two types of symbols at random, there will be $2^2 = 4$ distinct ways if the order matters: {aa, bb, ab, ba}. The minimum length of a circular arrangement…
Claire S.
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Product of consecutive transpositions.

I was interested if this equality holds for arbitrary $n$: $$(1 \space 2)(2\space3)(3\space 4)...(n-1\space n) (n\space 1)=(1)(2 \space 3 \space 4... \space n-1 \space n)$$ (considering multiplications are done in $S_n$) It looks like it is true…
Easy Cheat
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shortest factorization permutation has length n-c(π)

Prove that the number of cycles in a permutation is always changed by exactly 1 when you multiply it by a transposition! more precisely if $c(π)$ denote the number of cycle in the permutation $π$ and if $τ = (x y)$ is a transposition you are to…
Syeda Ssyed
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2 answers

>Let $ \sigma_{2} = (4215)(3426)(5617)$. Show that permutation as products of, in pair, disjoint cycles and as product of transpositions.

Let $ \sigma_{2} = (4215)(3426)(5617)$. Show that permutation as products of, in pair, disjoint cycles and as product of transpositions. I am a beginner and I am not sure how to start. Any hint helps! Thanks.
user560461
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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 proceeds by saying that $s_1 s_2 = [3\ 1\ 2]$ and…
virttop
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How to optimally distribute relative position using an $n$-cycle

Consider $n$ objects arranged in order: $A=\{1,2,\ldots,n\}$ and an $n$-cycle $\varphi$ from $S_n$. To motivate what I will ask, consider when $\varphi=(12\ldots n)$, and how the powers of $\varphi$ act on $A$ . For all powers $m$ (mod $n$),…
2'5 9'2
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Decomposition of permutation from disjoint cycle

I am a beginner, so my notations could be wrong. Consider in disjoint cycle notation: $ X = (1)(2, 3, 16, 8, 10, 5, 14, 4, 12, 13)(6, 9)(7)(11, 15) $, $Y = (1, 10, 6, 12, 8, 9, 3)(2, 14, 16, 11, 13, 15, 4, 7) (5)$, $Z = (1, 3, 14, 15, 16, 12, 11,…
hola
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Multiplying Cycles and permutation

So I did this task "Let A4 be the alternating group on 4 letters, that is the subgroup of S4 consisting of even permutations. Find elements σ1, σ2, σ3 ∈ A4 such that σ1 has order 1, σ2 has order 2, and σ3 has order 3." and got σ1 = e, σ2 = (1, 2)(3,…
user1162295
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Requirement for an even permutation to be a square

Given an even permutation $\sigma,$ what more is required for it to be the square of some other permutation? Is it enough that there be an even number of transpositions in its cycle decomposition? I looked at the case $\sigma=(12)(3456)$ and could…
coffeemath
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What is the effect of the composition $f ∘ g ∘ f^{−1}$ on g?

Let f = (4 5 6) and g = (1 9 8 4)(2 7 5)(3 6) be two permutations in S9. Then $f ◦ g ◦ f^{−1}$ = (1 9 8 5)(2 7 6)(3 4) What is the effect of such a composition on g? I can see 2 potential similarities. The first one is that the 4 5 and 6 have been…
Bill Cogn
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Permutation Cycles for work on algebra

What books can I use that focus on permutation cycles? I am working on algebra, particularly on work by Leibniz where he uses the permutation group Sn.
Veak
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Conjugated elements of $S_n$ for $n\geq 1.$

If $n\geq 5.$ Prove for all $(ab)(cde)$ of $S_n$ (with different $a,b,c,d,e$) are conjugated. My proof is the following: Given $\alpha=(a_1b_1)(c_1d_1e_1)$ y $\beta=(a_2b_2)(c_2d_2e_2)$,…
Cristhophel Saramango
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Inverse permutation

Can anyone help me with this question. ((14638)(259))^-1 I understand the normal way of multiplying permutation with cycle notation but here is an inverse sign so I don’t know what to do? Many thanks.
Chris
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