What are the inverses of the following elements of S4:
(a) (1234)
(b) (4321)
(c) (12)(34)
(d) (243)
(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).
