GATE CS 2010 Q4 — Monoids and Groups

To determine the algebraic structure of $(S, *)$, let us analyze the properties of the set $S = \{1, \omega, \omega^2\}$ under complex multiplication ($*$):

1. Closure: The multiplication table for $S$ is:

   $*$	$1$	$\omega$	$\omega^2$
   $1$	$1$	$\omega$	$\omega^2$
   $\omega$	$\omega$	$\omega^2$	$1$
   $\omega^2$	$\omega^2$	$1$	$\omega$

   Since all products belong to $S$, the closure property holds.
2. Associativity: Complex multiplication is associative. Thus, $(a * b) * c = a * (b * c)$ for all $a, b, c \in S$.
3. Identity Element: The element $1 \in S$ acts as the identity because $1 * x = x * 1 = x$ for all $x \in S$.
4. Inverse Element: Every element has a unique inverse in $S$:
   • The inverse of $1$ is $1$ (since $1 * 1 = 1$).
   • The inverse of $\omega$ is $\omega^2$ (since $\omega * \omega^2 = 1$).
   • The inverse of $\omega^2$ is $\omega$ (since $\omega^2 * \omega = 1$).

Since all four group axioms are satisfied, $(S, *)$ forms a Group.

Why other options are incorrect:

• A Ring, Integral Domain, and Field require two binary operations (typically addition and multiplication) defined on the set. Since only one operation ($*$) is defined here, these structures are not applicable.