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http://sporadic.stanford.edu/bump/group/gr1_4.html
Exercise 1.4.10: Show that in the decomposition of a permutation into a product of disjoint cycles, the cycles are unique (though their order is not, by Proposition 1.4.1). Now let $\sigma \in S_n$. We will explain how to associate with $\sigma$ a partition. We consider its decomposition into a product of disjoint …
https://math.stackexchange.com/questions/1819720/disjoint-permutations
Two permutations are disjoint if any point moved by one permutation is fixed by the other permutation. In other words, in the disjoint cycle decomposition of the two permutations, there is no overlap in the points that are written out. Equivalently, two permutations are disjoint if and only if they have disjoint support (the support of a ...
https://reference.wolfram.com/language/ref/PermutationSupport.html
The support of a permutation perm is the list of integers that are not fixed by perm. The list of integers is returned sorted. PermutationSupport works with Cycles objects as well as with permutation lists. When applied to a permutation list, PermutationSupport [{p 1, …, p n}] returns the p i for which p i ≠ i.
http://web.eecs.umich.edu/~valeria/research/thesis/thesis4.pdf
into the NOR of disjoint-support components: F "v w" f1 fn * (4.1) then, provided that no component function fi is itself the OR of other disjoint-support functions, the functions fi are uniquely determined, up to a permutation.
https://groupprops.subwiki.org/wiki/Cycle_decomposition_for_permutations
Definition For finite sets. Let be a set and be a permutation. A cycle decomposition for is an expression of as a product of disjoint cycles.. Here, a cycle is a permutation sending to for and to .Two cycles are disjoint if they do not have any common elements. Any permutation on a …
https://www.physicsforums.com/threads/if-two-permutations-commute-they-are-disjoint.365573/
Oct 16, 2011 · The other way round is easy to see, since if two cycles are disjoint they do not do anything with the numbers permuted by the other cycle, hence they commute. But I don't know how to start when I want to prove the statement above... can anyone hint me?
https://www.maths.tcd.ie/~levene/1214/pdf/prob3sol.pdf
Problem session solutions 1.Write the following permutations as a product of disjoint cycles: (a) 1 2 3 4 5 5 1 2 4 3 (b) 1 2 3 4 5 5 4 3 2 1 (c) 1 2 3 4 5 6 7 8
http://reference.wolfram.com/mathematica/tutorial/Permutations.html
Permutations are basic elements in algebra. They have a natural non-commutative product (as matrices do as well), and hence can encode highly nontrivial structures in a compact way. Permutations provide a way of representing any finite group, which makes them key tools in many applications in mathematics, science, engineering, or even art. In particular, permutations play a central role in the ...
https://reference.wolfram.com/language/ref/Cycles.html
Cycles must be disjoint, that is, they must have no common points. Cycles objects are automatically canonicalized by dropping empty and singleton cycles, rotating each cycle so that the smallest point appears first, and ordering cycles by the first point. Cycles [{}] represents the identity permutation.
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