Q. In the group $\left(G, \otimes_{15}\right)$, where $G=\{3,6,9,12\}, \otimes_{15}$ is multiplication modulo $15$, the identity element is
ManipalManipal 2008
Solution:
Given, $G=\{3,6,9,12\}$
The composition table is
$x_{15}$
$3$
$6$
$9$
$12$
$3$
$9$
$3$
$12$
$6$
$6$
$3$
$6$
$9$
$12$
$9$
$12$
$9$
$6$
$3$
$12$
$6$
$12$
$3$
$9$
From the table $6$ is the identity element.
| $x_{15}$ | $3$ | $6$ | $9$ | $12$ |
|---|---|---|---|---|
| $3$ | $9$ | $3$ | $12$ | $6$ |
| $6$ | $3$ | $6$ | $9$ | $12$ |
| $9$ | $12$ | $9$ | $6$ | $3$ |
| $12$ | $6$ | $12$ | $3$ | $9$ |