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### The Maximum possible order for an element $S_n$ [duplicate]

Given the following groups, what is the maximum possible order for an element for (a) $S_5$ (b) $S_6$ (c) $S_7$ (d) $S_{10}$ (e) $S_{15}$ My book justifies the answer as (a) The greatest ...
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### Order of cyclic subgroups in symmetric groups [duplicate]

What is the largest possible order of a cyclic subgroup of $S_7$? What's an example of this? I really just need a better understanding of cyclic subgroups of symmetric groups. I know that the ...
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### What is maximal possible order of an element in $S_{10}$ ? Why? [duplicate]

What is maximal possible order of an element in $S_{10}$? Why? Give an example of such an element. How many elements will there be in $S_{10}$ of that order?
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### Upper bound on the order of elements in the symmetric group [duplicate]

In Vinberg's textbook on algebra you are asked in an exercise to prove that the order of any element in the symmetric group on n letters does not exceed e^(n/e), which the book tells you is ...
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### What is the largest possible order of a permutation in $S_{n}$? [duplicate]

I was trying to solve the following question (from this link): What is the largest possible order of a permutation in $S_{10}$? I though there is some sophisticated way to solve it but then I ...
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### What's the smallest exponent to give the identity in $S_n$?

Let $S_n$ denote the symmetric group on $n$ letters. We know that $\tau^{n!} = e$ for any element $\tau \in S_n,$ where $e$ denotes the identity element. Can we find a smaller positive integer $m$ ...
3k views

### maximum order of an element in symmetric group [duplicate]

While doing my homework i find out that the maximum order of an element in $S_3$ is 3 (the element $(123)$) and the maximum order of an element in $S_4$ is 4 (the element $(1234)$) Can i generalize ...
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### How do I find the permutation with the highest order in a symmetric group?

My professor gives this text, but I don't understand what it's saying, could someone explain it to me? Let $M(n)$ denote the largest order of an element in $S_n$. By Theorem 1 $M(n)$ is the largest ...
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### Finding the inverse of an element of $S_n$ and it's order [duplicate]

I have two questions, 1) What are the ways to find the inverse of an element of $S_n$? 2) What are the ways to find the order of an element of $S_n$?
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### Is there a good way to show that the order of element in $S_7$ are at most $12$?

The only solution would be going through all cycle types of all permutations which is a lot of work. Is there any smarter solution than this one? Thank you in advance!
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### What is the maximum order element in $S_{13}$

I want to find the maximum possible order of an element in $S_{13}$. I have found that disjoint cycles of length $(2)(3)(7)$ give order 42, However ,answer in my book is given to be $60$ Can someone ...
### If P is an nxn permutation matrix, is there an upper bound on k such that $P^k = I$?
So if P is an nxn permutation matrix, then because there are only finitely many ways to permute finitely many elements, we know that the sequence $P, P^2, P^3, ...$ eventually has to repeat, between ...
Let $n$ be a natural number. How can I find a partition of $n$ such that its least common multiple is maximized? In other words, how can I find $a_1, a_2, \cdots a_m \in \mathbb{N}$ such that a_1 + ...