Question

If o = (1 2 3 4)(5) then order of o is

Answer 1

Answer:

The order of o is 4

Step-by-step explanation:

• o is an element of symmetric group $S_{5}$  i.e group of all the permutations on the set of five elements namely 1,2,3,4,5

• Order of an element of a group is defined as the least power (n) of the element (o) such that $o^{n} =e$ where 'e' is identity element of that particular group ( (1)(2)(3)(4)(5) is the identity in case of $S_{5}$ )
• To find the order of a permutation we use direct formula that is LCM of length of disjoint cycles.
• Disjoint cycles are those cycles that have no element in common like (1 2 3 4)(5)

Step 1

• First check if the given cycle is disjoint or not.
• If it is not then make it disjoint.
• But since all the elements appears only once in 'o' therefore the given cycle is disjoint

Step 2

• Now count the length of cycle
• There are 2 disjoint cycles i.e (1 2 3 4) and (5)
• Length of (1 2 3 4) is 4 and length of (5) is 1

Step 3

Take LCM of both the lengths of cycles obtained i.e LCM(4,1)

We get

LCM (4,1)=4

Therefore the order of o is 4