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Consider the elements of $S_7$. For each $\sigma \in S_7$ there is a smallest positive integer |$\sigma$| such that $\sigma^{|\sigma|}=e$. Find the value of $N$= max{ $|\sigma|$ | $\sigma \in S_n$}. Give a specific permutation $\alpha \in S_7$ such that $|\alpha|=N.$


So honestly I have no idea where to even start. Any and all guidance is greatly appreciated.

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    $\begingroup$ You could write out a representative for each possible disjoint cycle structure and calculate the order of each representative: eg (12)(34)(567) or (12)(34567) are two possibilities for the cycle structures, and the orders are 6 and 10 respectively for these examples. $\endgroup$ – Geoff Robinson Mar 9 '14 at 21:51
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Your question is equivalent to find nonnegative integers $a_1+a_2...+a_k=7$ such that $lcm(a_1,,,a_k)$ is maximum.

To make it maximum,you should choose $a_i$ relativly prime as much as you can.After some try,you can see that answer is $12$.($a_1=3$ and $a_2=4$)

To generalize the result for $S_n$ may be challinging.

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As mesel said, this is finding same as maximizing $lcm(a_1,..,a_k)$ for $a_i$ such that $a_1+a_2+...+a_k=7$. This follows from the fact that disjoint cycles commute and the order is then just the number of times a cycle must be composed with itself to make each of these disjoint cycles $e$.

But this is a hard question for general $n$- you know that once you have professional mathematicians working to find asymptotics it's a difficult problem.

But for $S_7$, we can see that the highest order is $12$ just by checking the conjugacy classes of n-cycles and fixed-points and finding the largest relatively prime integers. So the answer is $3 \times 4=12$ and we have $(1234)(567)$ as our $\sigma$.

To build some intuition/number sense behind this, we also see that $3$ and $4$ are the integers that are closest to equal and still fit the conditions of adding up to $7$ and being coprime, so their product must be greatest of combinations $a_i$.

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