Question

For $g\isin\Z$, let $\bar{g}\isin\Z_{37}$ denote the residue class of g modulo 37. Consider the group U37 = {$\bar{g}\isin\Z_{37}:1\le g\le37$ with gcd(g, 37) = 1} with respect to multiplication modulo 37. Then which one of the following is FALSE?

• A. The set ${\bar g \isin U_{37}: \bar g = (\bar g)^{-1}}$ contains exactly 2 elements.
• B. The order of the element $\overline{10}$ in U37 is 36.
• C. There is exactly one group homomorphism from U37 to ($\Z$, +).
• D. There is exactly one group homomorphism from U37 to ($\mathbb{Q}$,+).

Answer 1

Answer: (B)

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

Answer 2

The correct option is (B): The order of the element $\overline{10}$ in U37 is 36.