GATE CS 2004 Q72 — Monoids and Groups

To find the last row of the group table, we can analyze the algebraic structure of the given 4-element group $G = \{e, a, b, c\}$.

1. Identify the Identity Element

From the first row of the table, we have:

This confirms that $e$ is the identity element of the group.

2. Determine the Group Structure

From the second row of the table, we observe the powers of $a$:

• $a^1 = a$
• $a^2 = a * a = b$
• $a^3 = a * b = c$
• $a^4 = a * c = e$

Since $a$ generates all elements of the group, $G$ is a cyclic group of order $4$ generated by $a$, isomorphic to $(\mathbb{Z}_4, +_4)$ under the mapping:

3. Compute the Last Row (Row for $c$)

Using the cyclic relations ($c = a^3$, $b = a^2$, and $a^4 = e$), we compute the products for the row of $c$:

• $c * e = c$ (since $e$ is the identity)
• $c * a = a^3 * a = a^4 = e$
• $c * b = a^3 * a^2 = a^5 = a$
• $c * c = a^3 * a^3 = a^6 = a^2 = b$

Thus, the last row of the table is $c \; e \; a \; b$.