129 and K a Is H a normal subgroup of G b Is K a normal subg

(129) and K a) Is H a normal subgroup of G? b) Is K a normal subgroup of G? 2. Let G-S3, H (12)

Solution

(a) Since (1,2,3)(1,23)(1,2,3)=(1). Therefore H={(1), (1,2,3), (1,3,2)=(1,2,3)(1,2,3)}, Order of H=3 Order (G)=6. Therefore Index of H is O(G)/O(H)=2> Every Subgroup of Index 2 is normal. therefore H is normal.

(b) K is not normal . As (1,2)(1,2)=(1). Therefore K={(1), (1,2)}

Now, (1,3)K={(1,3),(1,2,3) and K(1,3)={(1,3),(1,3,2)}

Thus (1,3)K not equal to K(1,3). Thus K is not normal.