View source for GATE2010 q4 ← GATE2010 q4

Consider the set S = $\{1, ω, ω^2\}$, where $\omega$ and $\omega^2$ are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

(A) A Group

(B) A Ring

(C) An integral domain

(D) A field

Solution by Happy Mittal[edit]

We can directly answer this question as "A Group", because other three options require two operations over structure, but let us see whether $(S, *)$ satisfies group properties or not.

• Closure: If we multiply any two elements of $S$, we get one of three elements of $S$, so $S$ is closed over $*$.
• Associativity: multiplication operation is anyway associative.
• Identity element: $1$ is identity element of $S$.
• Inverse element: inverse of $1$ is $1$ because $1 * 1 = 1$, inverse of $ω$ is $ω^2$, because $ω * ω^2 = 1$. Also inverse of $ω^2$ is $ω$, because $ω^2 * ω = 1$

Thus, $S$ satisfies all $4$ properties of group, so it is a group. In fact, $S$ is an abelian group, because it also satisfies commutative property.
So, option (A) is correct.