# Inverse of Permutation Group

• Last Updated : 22 Feb, 2021

Inverse of Permutation Group-: If the product of two permutations is the identical permutation then each of them is called inverse of each other.

For Example-: The permutations are inverse of each other since their product is which is an identical permutation.

Example 1-: Find the inverse of permutation Solution-: Let the inverse of permutation be \

where a, b, c and d are to be calculated.

Then According to definition of Inverse of Permutation or ∴ b=4 , c=2 , a=1 , d=3

∴ Required inverse is Example 2-: Calculate A-1 if A= Solution-: Let the inverse of A be where a, b, c, d and e are to be calculated.

Then According to definition of Inverse of Permutation or ∴ b=1 , c=2 , a=3 , e=4 , d=5

∴ We have A-1 Example 3-:  If then compute f-1o g-1.

Solution-:

f-1= g-1= f-1o g-1= f-1o g-1= Example 4-: If P1= , P2= ,P3= Find (P1 o P2)-1  and (P2 o P3)-1.

Solution-: P1 o P2= P2 o P3= Also, we know that if P-1 be the inverse of permutation P, then P-1 o P = I .

∴ (P1 o P2)-1 = inverse of ∴ (P2 o P3)-1 = inverse of Example 5-: Prove that (1  2  3  …….  n )-1 = ( n  n-1  n-3 …..  2  1)

Solution-: ( 1  2  3  …..  n)= = = = =I

Hence, (1  2  3  …….  n )-1 = ( n  n-1  n-3 …..  2  1)

My Personal Notes arrow_drop_up