Show that the subgroup generated by 123 is a normal subgroup

Show that the subgroup generated by (123) is a normal subgroup of S₃

Solution

Answer :

This element has order 3, being a cycle of length 3. Hence it is a subgroup of index 2.

(This follows from the proof of Lagrange’s Theorem:| S₃ | / < ( 1 2 3 ) > | = 6/3 = 2 and any subgroup ofindex 2 is normal. Alternatively, the group generated by ( 1 2 3 ) is A₃ ,so it’s the kernel of a group homomorphism.