Question

Which one of the following is TRUE for the symmetric group $S_{13}$ ?

1. $S_{13}$ has an element of order 42
2. $S_{13}$ has no element of order 35
3. $S_{13}$ has an element of order 27
4. $S_{13}$ has no element of order 60

Answer: $S_{13}$ has an element of order 42

Solution :-

Option (1)

This option is correct because $S_{13}$ has a Cycle Decomposition <6, 7>. So, the order of this Cycle Decomposition is 42, because l.c.m. of 6, 7 is 42 and addition of 6, 7 is 13.
So, $S_{13}$ has an element of order 42.

Option (2)

This option is wrong because $S_{13}$ has a Cycle Decomposition <1, 5, 7>. So, the order of this Cycle Decomposition is 35, because l.c.m. of 1, 5, 7 is 35 and addition of 1, 5, 7 is 13.
So, $S_{13}$ has an element of order 35.

Option (3)

This option is also wrong because $S_{13}$ has no any Cycle Decomposition whose order is 27 and addition of elements in that Cycle Decomposition is 13.
So, $S_{13}$ has no element of order 27.

Option (4)

This option is also wrong because $S_{13}$ has a Cycle Decomposition <1, 3, 4, 5>. So, the order of this Cycle Decomposition is 60, because l.c.m. of 1, 3, 4, 5 is 60 and addition of 1, 3, 4, 5 is 13.
So, $S_{13}$ has an element of order 60.

So, the correct answer is option-1, $\to S_{13}$ has an element of order 42