1 $\endgroup$ add a comment | 3 $\begingroup$ When a permutation is written as a product of disjoint cycles, its order is the least common multiple of those cycles' lengths (easy proof by induction). Codeforces. → Pay attention Before contest Kotlin Heroes: Practice 4 3 days Register now » The Order Theorem for Permutations. I know the order of a group is simply the number of elements in the group but I can't find any info on what the order of the above mentioned perm./cycles. ): (1, 2, 3) (4, 5, 6, 7, 8) (9, 10) ==> Order lcm (3, 5, 2) = 30. We define the $|$ order $|$ of a permutation written as the product of disjoint cycles to be the least common multiple $(\operatorname{lcm})$ of the lengths of those cycles. This question is regarding modern/abstract algebra. Permutation Groups . THE EXPECTED ORDER OF A RANDOM PERMUTATION WILLIAM M. Y. how do you find the length? Let Sbe a set. it is the lcm of the lengths if in cycle notation (which they are).
How do you find the order of a permutation? Idea of the proof: The set {1,2,...,n} splits into 3 subsets: elements moved by π, elements moved by σ, and elements ﬁxed by both π and σ. THE EXPECTED ORDER OF A RANDOM PERMUTATION WILLIAM M. Y. Now we can address the order of a permutation: We define the length of a cycle to be the number of elements in the cycle. Order of a particular given permutation = LCM(order of all disjoint cycles) ? Lemma 5.3. $$\operatorname{order}(123)(45678) = \operatorname{lcm}(3, 5) = 15.$$ share | cite | improve this answer | follow | | | | edited Sep 22 '14 at 12:28. answered Sep 22 '14 at 12:21. amWhy amWhy. It turns out that the maximal order is 30. The Order of a Permutation.
Abstract Algebra/Group Theory/Permutation groups. Give an example that G has not an element whose order is the least common multiple of m and n. Hot Network Questions History of non-American software/hardware/CS theory development, 1940s-1980s? Permutation groups De nition 5.1. Definition: If is a permutation of the elements in then the order of denoted is the smallest positive integer such that where is the identity permutation. Well-known. Recall from The Order of a Permutation page that if $\sigma$ is a permutation of the elements in $\{1, 2, ..., n \}$ then the order of $\sigma$ is the smallest positive … The smallest possible integer that is the LCM of lengths of the cycle of permutation is known as the order of permutation.