For g ∈ Z, let g̅ ∈ Z₃₇ denote the residue class of g modulo 37. Consider the group U₃₇ = {g̅ ∈ Z₃₇ ∶ 1 ≤ g ≤ 37 with gcd(g, 37) = 1} with respect to multiplication modulo 37. Then which one of the following is FALSE?

The set {g̅ ∈ U₃₇ ∶ g̅ = (g̅)⁻¹} contains exactly 2 elements.

The order of the element \(\overline{10}\) in U₃₇ is 36.

There is exactly one group homomorphism from U₃₇ to (Z, +).

There is exactly one group homomorphism from U₃₇ to (Q, +).

For g ∈ Z, let g̅ ∈ Z37 denote the residue class of g modulo 37. Consider the group U37 = {g̅ ∈ Z37 ∶ 1 ≤ g ≤ 37 with gcd(g, 37) = 1} with respect to multiplication modulo 37. Then which one of the following is FALSE?