Question

Let $\mathbf{G}\mathbf{=}\mathbf{\{}\mathbf{e}\mathbf{,}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{\}}$ be a group of order 3. Write out the operation table for G

Short answer

The operation for the group G can be written as follows:

$\mathbf{*}$	e	a	b
e	e	a	b
a	a	b	e
b	b	e	a

Step-by-step solution

Exercise 28 of section 7.1

Each element of a finite group G appears exactly once in each row and exactly once in each column of the operation table.

Operation table for G

Consider the group $G=\{e,a,b\}$ of order 3.

Since, the product of any element with the identity is the element and then by exercise 28 of section 7.1, the operation table can be made as follows:

$*$	e	a	b
e	e	a	b
a	a	b	e
b	b	e	a

Therefore, the operation of G is shown above.