Consider the group homomorphism ? : S3 ×S5? S5 and ?((?, ? )) = ? .
(a) Determine the kernel of ?. Prove your answer. Call K thekernel.
(b) What are all the left cosets of K in S3×S5 using set builder notation.
(c) What are all the right cosets of K in S3 ×S5 using set builder notation.
(d) What is the preimage of an element ? ? S5 under??
(e) Compare your answers in parts (b)-(d). What do younotice?
