Question

If $\mathit{n}\mathbf{\ge }\mathbf{5}$, what is the center of${\mathit{A}}_{\mathbf{n}}$ ?

Short answer

The center of $A_n$for$n\ge 5$ isrole="math" localid="1654604878170" $\left\{\mathrm{e}\right\}$ .

Step-by-step solution

Determine the center of An

Consider $n\ge 5$.

Starting the point out that for any$n\ge 5,A_n$ is simple.

Further, the center of any group is normal subgroup.

Result

Consequently, the center of $A_n$for $n\ge 5$ is either $\left\{\mathrm{e}\right\}$or $A_n$itself.

As $A_n$is not abelian, the center of $A_n$is not abelian.

Thus, the center of $A_n$for$n\ge 5$ is$\left\{\mathrm{e}\right\}$ .