what is the maximom order of an element is $\mathbb S_{15}$

Question

Let $S_n$ be the permutation group of order $n$. What is the maximom order of an element on $\mathbb S_{15}$.

Is there any way to find maximom order of an element in $\mathbb S_{15}$ or find the order of an arbitary element in $\mathbb A_n$ and $\mathbb S_n$

Hint: What are the possible cycle structures of elements in $\mathbb{S}_{15}$? Using that for disjoint cycles $\sigma_i$ we have $\left|\prod_i \sigma_i\right| = \text{lcm}(|\sigma_1|, \ldots |\sigma_k|)$ gives the order of all elements of a certain cycle type. — Travis Willse, Jan 29 '15 at 12:04

score 1 · Accepted Answer · answered Jan 29 '15 at 12:05

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The order of an element $g$ in $S_n$ is the LCM of the length of cycles in the factorizetion of $g$ into disjoint cycles. From here it is a question in number theory. I thing for $n=15$ the answer $105=7\cdot 5\cdot 3$.

answered Jan 29 '15 at 12:05

Ofir Schnabel

3,891

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$105$ is indeed correct (as confirmed by a quick computation in magma). – Sebastian Schoennenbeck Jan 29 '15 at 12:07
What is $g$ in your text – erfan soheil Jan 29 '15 at 12:35
Interested in the same question, can you please explain how did you get $105=7\cdot5\cdot3$ ? – grayQuant Feb 10 '15 at 17:11
@grayQuant I am not sure I understand your question. $105$ is divisble by $3$, divide and get $105=3\cdot 35$. Since $35=5\cdot 7$ we get $105=3\cdot 5\cdot 7$. – Ofir Schnabel Feb 11 '15 at 08:07
How did you get 105? – grayQuant Feb 11 '15 at 15:42
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@grayQuant we looking for elements in $S_{15}$. then we looking on length of cycles $n_1,n_2,\ldots ,n_r$ such that $n_1+n_2+\ldots +n_r\leq 15$ and LCM$(n_1,n_2,\ldots ,n_r)$ is maximal. I found that $3,5,7$ is the best possibility. – Ofir Schnabel Feb 11 '15 at 15:49
So for $S_14$, is it $2(5)(7)=70$? – grayQuant Feb 11 '15 at 16:18
@grayQuant I think in $S_{14}$ it is $3\cdot 4\cdot 7=84$. There is no general solution to this problem. – Ofir Schnabel Feb 12 '15 at 08:09

what is the maximom order of an element is $\mathbb S_{15}$

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