GATE2010 q4 Revision as of 07:11, 15 April 2014 by Arjun Suresh (talk | contribs) (diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Consider the set S = $\{1, ω, ω^2\}$, where $ω$ and $ω^2$ 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

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.

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