Groups Questions and Answers

The set of positive rational numbers forms a group under the operation $a * b = \frac{ab}{10}$. Find the identity element.

Under the operation $a * b = \frac{ab}{10}$ on the positive rational numbers, write the inverse of an element $a$ in terms of $a$.

The positive rational numbers form a group under $a * b = \frac{ab}{10}$. Which group axiom fails if the set is narrowed to the positive integers?

The set $G = \{0, 1, 2\}$ forms a group under addition modulo 3. State the identity element.

In the group $G = \{0, 1, 2\}$ under addition modulo 3, find the inverse of the element 2.

The set $G = \{0, 1, 2\}$ under addition modulo 3 is shown to be abelian. Which observation establishes that?