Find all subgroups of S3 and determine which ones are normal

Find all subgroups of S3 and determine which ones are normal.

Solution

. Lagrangeís theorem implies that the subgroups of S3 = Aut(fa; b; cg) are of orders 6; 3; 2; 1: A direct veriÖcation shows that they are S3; A3 = {(a b c);(a c b); id}; {(a b); id}; {(a c); id}; {(b c); id}; {id}: A direct calculation shows that the normal subgroups are {id}; A3; S3: