Q 4. In this question we will consider the group G = (U40, ⊗40), that is, the integers with a multiplicative inverse modulo 40. We aim to discover a product of cyclic groups that is isomorphic to G, as guaranteed by the classification of finite Abelian groups.

(a) Write down the elements of G as a set. What is the order of G?

(b) Calculate h3i, the cyclic subgroup of G generated by 3.

(c) Based on (b), which of the following groups could not be isomorphic to G? Why?

Z2 × Z2 × Z2 × Z2 Z2 × Z2 × Z4 Z2 × Z8 Z4 × Z4 Z16

(d) How many elements of order 2 are there in Z16? How about Z4 × Z4 and Z2 × Z8?

(e) If an element a has order 2 then a 2 = 40b + 1 for some integer b. Find four elements of order 2 by solving this equation using b = 2 and b = 3.

(f) Which of the groups listed in part (c) is isomorphic to G? Explain your answer.