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- Cycles_and_fixed_points abstract "In mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S. These orbits are subsets of S that can be written as { c1, ..., cl }, such thatπ(ci) = ci + 1 for i = 1, ..., l − 1, and π(cl) = c1.The corresponding cycle of π is written as ( c1 c2 ... cn ); this expression is not unique since c1 can be chosen to be any element of the orbit.The size l of the orbit is called the length of the corresponding cycle; when l = 1, the single element in the orbit is called a fixed point of the permutation. A permutation is determined by giving an expression for each of its cycles, and one notation for permutations consist of writing such expressions one after another in some order. For example, letbe a permutation that maps 1 to 2, 6 to 8, etc. Then one may writeπ = ( 1 2 4 3 ) ( 5 ) ( 6 8 ) (7) = (7) ( 1 2 4 3 ) ( 6 8 ) ( 5 ) = ( 4 3 1 2 ) ( 8 6 ) ( 5 ) (7) = ...Here 5 an 7 are fixed points of π, since π(5)=5 and π(7)=7. It is typical, but not necessary, to not write the cycles of length one in such an expression. Thus, π = (1 2 4 3)(6 8), would be an appropriate way to express this permutation.There are different ways to write a permutation as a list of its cycles, but the number of cycles and their contents are given by the partition of S into orbits, and these are therefore the same for all such expressions.".
- Cycles_and_fixed_points thumbnail Gray_code_*_bit_reversal_16.svg?width=300.
- Cycles_and_fixed_points wikiPageID "2221032".
- Cycles_and_fixed_points wikiPageRevisionID "603761118".
- Cycles_and_fixed_points hasPhotoCollection Cycles_and_fixed_points.
- Cycles_and_fixed_points subject Category:Fixed_points_(mathematics).
- Cycles_and_fixed_points subject Category:Permutations.
- Cycles_and_fixed_points comment "In mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S. These orbits are subsets of S that can be written as { c1, ..., cl }, such thatπ(ci) = ci + 1 for i = 1, ..., l − 1, and π(cl) = c1.The corresponding cycle of π is written as ( c1 c2 ...".
- Cycles_and_fixed_points label "Cycles and fixed points".
- Cycles_and_fixed_points sameAs m.06w_np.
- Cycles_and_fixed_points sameAs Q5198198.
- Cycles_and_fixed_points sameAs Q5198198.
- Cycles_and_fixed_points wasDerivedFrom Cycles_and_fixed_points?oldid=603761118.
- Cycles_and_fixed_points depiction Gray_code_*_bit_reversal_16.svg.
- Cycles_and_fixed_points isPrimaryTopicOf Cycles_and_fixed_points.
