Finding element of maximal order in symmetric group

**jackie** · December 13th 2009, 09:26 PM

Would someone show me the method to find an element of maximal order in $S_7$ ? Is there a general technique for this?
I think that this element could have order 12=3 $\times$ 4 since I can find a (4,3) cycle element in $S_7$ like (1234)(567) which has order 12.

**NonCommAlg** · December 13th 2009, 10:20 PM

Originally Posted by jackie

Would someone show me the method to find an element of maximal order in $S_7$ ? Is there a general technique for this?
I think that this element could have order 12=3 $\times$ 4 since I can find a (4,3) cycle element in $S_7$ like (1234)(567) which has order 12.

$12$ is the answer. if there exists an integer $m=p_1^{k_1}p_2^{k_2} \cdots p_r^{k_r},$ where $p_1, \cdots, p_r$ are distinct primes, such that $p_1^{k_1} + p_2^{k_2} + \cdots + p_r^{k_r} = n,$ then the maximal order of elements in $S_n$ is $m.$

for example, since $12=2^2 \times 3$ and $2^2 + 3 = 7,$ the maximal order of elements in $S_7$ is $12.$

see section 4 (the prime connection) of this paper for more details. don't get scared, it's very easy to understand!

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