What are the inverses of the following elements of S4:

(a) (1234)

(b) (4321)

(c) (12)(34)

(d) (243)

Guest Jun 13, 2023

#1**0 **

(a) (1234): To find its inverse, we need to find a permutation such that when composed with (1234), it returns the identity permutation. We can observe that the inverse of (1234) is (1432), as their composition yields the identity permutation:

(1234) * (1432) = (1)(2)(3)(4) = e,

where e represents the identity permutation.

Therefore, the inverse of (1234) is (1432).

(b) (4321): To find its inverse, we need to find a permutation such that when composed with (4321), it returns the identity permutation. We can observe that the inverse of (4321) is (4321) itself since they are the same permutation:

(4321) * (4321) = (4)(3)(2)(1) = e.

Therefore, the inverse of (4321) is (4321).

Guest Jun 13, 2023