## Find all possible orders of elements in Sym(5)

Find all numbers $n$ such that $S_5$ contains an element of order $n$.

Note that the only possible cycle decompositions for elements in $S_5$ (up to the order of the cycles) are $1$, $(- -)$, $(- - -)$, $(- -)(- -)$, $(- -)(- - -)$, $(- - - -)$, and $(- - - - -)$. By a previous theorem, the order of a product of disjoint cycles is the least common multiple of the lengths of the factors. Thus, the possible orders for elements in $S_5$ are (in increasing order) 1, 2, 3, 4, 5, and 6.

• rip  On February 5, 2011 at 3:23 pm

Hi,

Perhaps you want to change “Sym(6)” in the title to “Sym(5)” ?

• nbloomf  On February 5, 2011 at 3:43 pm

Ahem… yes. 🙂

Thanks!

• kimochis  On October 5, 2013 at 2:28 pm

why is order 4 possible?
in case (–) (–) 1, the order is 2

• kimochis  On October 5, 2013 at 2:29 pm

why is order 4 possible?
in case (a, b ) (c, d) 1, the order is 2

• kimochis  On October 5, 2013 at 2:31 pm

Sorry, I was wrong, order 4 is possible in the case (a,b,c,d) 1