# 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)

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