Question

Find the center of the group ${\mathit{A}}_{\mathbf{4}}$

Short answer

The center of alternating group$A_4$ is $Z\left(A_4\right)=\left\{\left(1\right)\right\}$.

Step-by-step solution

Center of a group

If G is a group then the center of the group$\mathit{Z}\mathbf{\left(}\mathbf{G}\mathbf{\right)}$ is the set of elements that commute with all the elements in G.

Find the center of the group A4

The center of the group is represented by,

$Z\left(G\right)=\{x\in G|xg=gx,g\in G\}$

It is known that the symmetric group contains $n!$elements.

The symmetric group $S_4$has $24$permutation, and alternating group $A_4$has $12$permutation. There is only identity which commutes with the element of group $A_4$. Then,$Z\left(A_4\right)=\{x\in A_4|\left(123\right)\left(1\right)=\left(1\right)\left(123\right),\left(123\right)\in A_4\}$

Therefore, the center of alternating group$A_4$ is $Z\left(A_4\right)=\left\{\left(1\right)\right\}$.