The Maximum possible order for an element $S_n$
网页Given the following groups, what is the maximum possible order for an element for. (a) S5 S 5 (b) S6 S 6 (c) S7 S 7 (d) S10 S 10 (e) S15 S 15. My book justifies the answer as. (a) The greatest order is 6 6 and comes from a product of disjoint cycles of length 2 and 3. (b) The greatest order is 6 6 and comes from a cycle of length 6 6.
What are the elements of order $n$ in symmetric group $S_n$?
网页2013年8月29日 · Maybe but title explains it nicely, but it is customary to include the question in the body, not just the title. – Thomas Andrews. Aug 29, 2013 at 18:43. Add a comment. Sn S n. n n n n (n − 1)! (n − 1)! n n Sn S n. n = 6 n = 6 n = 10 n = 10 n = 12 n = 12.
Finding the inverse of an element of $S_n$ and it's order
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abstract algebra - $S_n$ is a group - Mathematics Stack Exchange
网页2020年3月9日 · $\begingroup$ "There are four requirements we need to verify". I believe the things went historically the other way around: after having realized that the set of bijections on a set does fulfill the four properties, where "Inverses" is actually the qualifying one (e.g. associativity holds for any map on a set), someone thought to define "group" an algebric …
Number of elements of each order in Sn [duplicate]
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Prove that the order of an element in - Mathematics Stack Exchange
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Proving that $S_n$ has order $n!$ - Mathematics Stack Exchange
网页Here's what I'd do: identify Sn with the subgroup of Sn + 1 consisting of those permutations that fix n + 1. Choose elements σ1, …, σn + 1 of Sn + 1 such that σi(n + 1) = i. For example, you could take σi to be [i, n + 1]. Then prove that these are …
Centralizer of a given element in - Mathematics Stack Exchange
网页Answer: k ⋅ (n − k)! is the number of elements in Tτ and this is so straightforward that you should be able to show this yourself. Step 3: Finally notice that Tτ defined as above is a subset of the centralizer Cτ of τ. All these would tell you that Cτ is in fact isomorphic (as a group) to Zk × Sn − k. Share.
Prove that the symmetric group on - Mathematics Stack Exchange
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abstract algebra - Find the center of the symmetry group $S_n ...
网页Here is a different proof based on the fact that the center of any group is precisely the set of elements whose conjugacy classes are singletons.