This is for abstract algebra class. Please show me all the details from scratch. Thank you.

2. Let G = Z x Zg. (a) Which elements of G have finite order. Justify your answer. (b) Verify that [G : ((1,1))] = 3. Explain why this implies that G/((1, 1)) ~ Z3. Hint: Note that G is not a finite group, so Lagrange's Theorem is not useful here. Instead, just identify all distinct left cosets of ((1, 1)) in G