The groups D6 R0 R60 R120 R180 R240 R300 F1 F2 F4 F5 F6 A4

The groups D_6 = {R_0, R_60, R_120, R_180, R_240, R_300, F_1, F_2, F_4, F_5, F_6} A_4 = {e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23)} are two groups of order 12. Explain why D_6 is not isomorphic to A_4.

Solution

D₆ has an element of order 6 , for example R₆₀ (this represents rotation by 60 degrees and hence its order is 6).

In A₄, elements have order 3 (all 3 cycles have order 3) or 2 (products of disjoint transpositions have order 2--the last three elements in the list).

Any isomorphism of groups preserves the orders .

As there is no element of order 6 in A₄, it follows that these two groups are not isomorphic.