List the elements of ZS3SolutionThe group S 3 contains one n

Solution

The group S 3 contains one normal subgroup Z3 generated by (1, 2, 3). It also contains 3 subgroups of order two < (12) >, < (13) >, < (23) >. We can use them to construct subgroups P3 = Z3 × Z3 S 3 × S 3 of oder 9 and P2 = Z2 × Z2 S 3 × S 3. The order of S 3 × S 3 is 22 × 3 2 . Thus P2, P3 are Sylow subgroups. The group P3 is normal, therefore it is the only 3-subgroup. n2 = 1 + 2k, n2 36/4 = 9 and n2|9. Thus n2 = 1, 3, 9. Combining different Z2 S 3 we obtain 9 subgroups in S 3 × S 3 of order 4. Thus n2 = 9 and our list is complete.