GATE CS 2006 Q3 — Monoids and Groups

To determine which statement is false, let us analyze the properties of the set $S = \{1, 2, 3, 5, 7, 8, 9\}$ under multiplication modulo $10$ ($\times_{10}$). The identity element of this system is $1$.

• Option A (It is not closed): This is a true statement. For example, $2 \times_{10} 5 = 10 \equiv 0 \pmod{10}$, but $0 \notin S$.
• Option B ($2$ does not have an inverse): This is a true statement. Any multiple of $2$ modulo $10$ is even or $0$, so we can never have $2 \times_{10} x = 1$ for any $x \in S$.
• Option D ($8$ does not have an inverse): This is a true statement. Similarly, any multiple of $8$ modulo $10$ is even or $0$, so $8$ cannot have an inverse.
• Option C ($3$ does not have an inverse): This is a false statement. Since $3 \times_{10} 7 = 21 \equiv 1 \pmod{10}$ and $7 \in S$, the inverse of $3$ exists and is $7$.

Since the question asks for the false statement, the correct option is C.