Question

Write a word problem about equal groups


and act it out to solve.

Answer 1

One scientist was working two molecules:

`BF_(3), SO_(3)`

And verified that geometrically speaking, those are trigonal planar molecules made up of an equilateral triangle. He designed both molecules and classified them as Plane dihedral Groups. Was he correct? Why?

Explanation:

Yes He was right. Place one atom in the centroid of the equilateral triangle (S or B), and each atom in each vertex(O or F).

We can apply geometric transformations (plane) rotations: R0, R2π/3, R4π/3

(And spatial ones, as R1, R2, R3 180º rotation over each median.)

Composite functions show that this is not an Abelian Group:

`R_(1)\circ\:R_(2)=R_{(2\pi)/(3)}\\R_(2)\circ\:R_(1)=R_{(4\pi)/(3)}\\`