Final answer:
In a dihedral group Dn, where n is the number of sides in an equilateral triangle, we can choose α as a 120 degrees counterclockwise rotation and β as a reflection about an axis passing through one of the vertices and the center of the triangle. Both α and β have order 3, satisfying the given conditions.
Step-by-step explanation:
To find group elements α and β where |α| = 3, |β| = 3, and |αβ| = 5, we can consider the elements in a dihedral group Dn. Let's take n = 3, which corresponds to the equilateral triangle.
In this case, α can represent a rotation of 120 degrees counterclockwise, and β can represent a reflection about an axis passing through one of the vertices and the center of the triangle.
Both α and β have order 3, which means that applying the operation three times will bring the elements back to their original state. Therefore, α and β satisfy the given conditions.