Group Visualization with Group Explorer - Quotient Groups in Multiplication Tables

Group Explorer allows us to carry out the same steps in the multiplication table view that we have done with Cayley diagrams. By highlighting the right cosets of the subgroup H  =  <(0 1)(2 3), (0 2)(1 3)> in the multiplication table, we get Figure 19, where the cosets are distinguished by color. It is apparent that if we view each coset as an individual unit, we are looking at a multiplication table for the group Z₃.

Figure 19. Multiplication table of A₄ showing the quotient by V₄

If we tried the same thing with a non-normal subgroup, say the subgroup generated by the permutation (0 1 2), we would get something like Figure 20. It is clear from the lack of order in Figure 20 that there is a significant difference as compared to Figure 19. One cannot step back and mentally cluster Figure 20 by color and expect to obtain a valid multiplication table for a group of order 4; the colors are too mixed to permit this.

Figure 20. Multiplication table of A₄ organized by a non-normal subgroup