Question

Give an example of a group in which the equation$x^2=e$ has more than two solutions.

Short answer

Additive group${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ is the group in which$x^2=e$ has more than two solutions.

Step-by-step solution

Solution of equation belonging to group

In a group $(G,\ast )$, if any element belonging to group $G$satisfies

$a\ast a=e$, then it is the solution of $x^2=e$.

Group with binary operation

Let us consider the group ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$with binary operation addition.

We know,$a=(0,0),b=(1,1),c=(0,1)\in {\mathbb{Z}}_2\times {\mathbb{Z}}_2$ .

Now,$\begin{array}{l}(0,0)+(0,0)=(0,0)\\ (1,1)+(1,1)=(0,0)\\ (0,1)+(0,1)=(0,0)\end{array}$

Therefore, there are three different elements in${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ which satisfies

$a+a=b+b=c+c=(0,0)$