Find a noncyclic subgroup of order 4 in the group U40 Prove

Solution

The elements of U(40) = {1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39}

Here, we have to find find two distinct elements of order 2.

Since 9² = 81 mod 40 = 1 mod 40. and -1 = 39 mod 40. and 39² = 1521 mod 40 = 1 mod 40. Thus, 9 and 39 are of order 2. Therefore, {9, 39} will be non-cyclic subgroup of order 4.

Further, we have |11| = |29| = 2 and (11.29) mod 40 = 319 mod 40 = 39. Since U(40) is abelian, the subset S = {1, 11, 29, 39} of U (40) is a subgroup of order 4. It is noncyclic because none of its elements have order 4.