The question is: How many elements in $S_4$ (the symmetric group of (1,2,3,4)) have order 6?
How can I think here?
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1categorize the elements in $S_4$ with respect to their cycle structure. – Colver Dec 10 '23 at 10:00
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This has been shown many times here. See this post for solving this in general. Also, please use MathJax. Here is a tutorial. – Dietrich Burde Dec 10 '23 at 10:02
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1Consider the classical decomposition into disjoint cycles : there are 3 types $(abcd), (a)(bcd),(ab)(cd)$ ; take exponents 2 or 3 on each , it will be difficult to find an order 6 ! – Jean Marie Dec 10 '23 at 10:04
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Using the Landau function $g(n)$ we have $g(4)=4$, so the maximal order of an element in $S_4$ is $4$. Hence $6$ is impossible. See this duplicate. – Dietrich Burde Dec 10 '23 at 10:07
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1@JeanMarie $4$ types, don't forget the identity. – jjagmath Dec 10 '23 at 10:07
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@jjagmath You are right ! – Jean Marie Dec 10 '23 at 10:09
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@JeanMarie Actually $5$. There's also $(ab)(c)(d)$ – jjagmath Dec 10 '23 at 10:12
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@jjagmath I am not well awaked this morning ! – Jean Marie Dec 10 '23 at 10:23
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I used the method proposed and got 4 but that is wrong. 5 is also wrong answer. – Kei Len Dec 10 '23 at 10:35
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1We are talking about the posible possible types of the decomposition into disjoint cycles (there are $5$) which you're supposed to use to find the number of elements of order $6$ which are zero. – jjagmath Dec 10 '23 at 13:22