Q. In the group $(G \bigotimes_{15}) $ where $G =\{3,6,9,12\},\bigotimes_{15}$ is multiplication modulo $15$, the identity element is
Solution:
Given, $G=\{3,6,9,12\}$
The composition table is
$x_5$
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.
| $x_5$ | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| 3 | 9 | 3 | 12 | 6 |
| 6 | 3 | 6 | 9 | 12 |
| 9 | 12 | 9 | 6 | 3 |
| 12 | 6 | 12 | 3 | 9 |