Question

Give an example of an abelian group of order 4 in which every non identity element a satisfies $\mathit{a}\mathbf{\ast }\mathit{a}\mathbf{=}\mathit{e}$ .

Short answer

${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$is an example of abelian group of order 4.

Step-by-step solution

Theorem 7.4

Let G and H be groups. Define an operation $•$on $G\times H$ by $(g,h)•(g^{\text{'}},h^{\text{'}})=(g\ast g^{\text{'}},h\circ h^{\text{'}})$then $G\times H$ is a group. If G and H are abelian, then $G\times H$ is also abelian. If G and H are finite, then$G\times H$ is finite and$|G\times H|=\left|G\right|\left|H\right|$ .

Example of abelian group 

Consider the abelian group ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$of order 4 with the elements $\left\{(0,0)(1,0)(0,1)(1,1)\right\}$ where $(0,0)$is an identity.

Let $(0,0),(1,0)\in {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$ then,

role="math" localid="1654251545264" $\begin{array}{c}(0,0)•(1,0)=(0\ast 1,0\circ 0)\\ =(0,0)\\ =\mathrm{e}\end{array}$

Similarly, for any two elements in ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$the product is the identity.

Therefore, ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$ is an example of order 4 in which every non-identity element a satisfies $\mathrm{a}\ast \mathrm{a}=\mathrm{e}$.