Question

Abstract Algebra

10.1.1. Find all normal subgroups of Ds and of S3.

Answer 1

10.1.1 find Page-1 all normal subgroups of Do and of S3.. Normal Subgroups of Do H- Reflection through line H V- Reflection through linev D- Reflection through line ol- Reflection through line o! Ro-oo- anticlockwise rotation R90° 90° - anticlock wise rotation R100 - 1000 -anticlock wise rotation R270° - 270° anticlock wise rotation Do = {Ro Rao, Rioo, R270 H, V, D, O'} {Ro} and Do are non trivial Subgroups of Do and hence {Ro} and Do normal subgroups of Do. Non-trivial Subgroups of Do are of orders 2 or 4. Subgroups of order a are ki = {Ro Rido} , K₂ = {Ro, v?, K3 = &RO,HZ, Ru= &RO,D3 K5 = {Ro,o'l. Subgroups of order u are K6 = {Ro, Rao , R 180 1 R270 }; ky = {Ro, R180 HV}; kB = {Ro, Rigo 10 de Since k, = 2(00); it is a normal Subgroup of Do. Rqo Ka Rao = {Ro Rgo vRao? = {Ro, H3 & 12 Hence K2 is not a normal Subgroup of Do Rao Kg Rad = {Ro Rqo H Rad} = {Ro,v3 & 13 Hence K₂ is not a normal subgroup of oo. LIFE WITH WITUS

Page 2 Rqo Ku Rqo = { Ro, Rgo DR 3 = &Ro, D'} & Ky Hence ky is not normal Subgroup of Da. Rqo ks Rao & ho, Rao D' Rao' } = {2o,0z & K5 Hence kes is not normal Subgroup of Do. Index of koky and ko in do is a Henee Subgroup of Index 2 is normal Hence K6, ky and ko are normal Subgroups. So Normal Subgroups of Do are {R} Do, ki , Kork, and ko. Normal Subgroups of S3 S3 = 3(1), (12), (13), (23), (123), (132)? trivial Subgroups of S3 are {(173 and Sz. Hence {(1) and So are normal in Sz. Non trivial Subgroups of Sy are of orders 2 and 3. Subgroups of order 2 k, = {(1), (12), K2 = {(1), (13)} K3 = {(1), (23)} Subgroups of order 3: Ku= {(1), (123), (132)} (23) K, (23) = {(1), (13)} & ki Hence ki is not a normal Subgroup. (23) K2 (235' = {(1), (1213 kg Hence K2 is not a normal subgroup.

Page - (12) K3 (125' = {(1), (13)} & K3 Hence K3 is not a normal subgroup of free. Sa har ky is a Subgroup of Index 2 in S3 Hence ky is normal subgroup of res. S3 So Normal Subgroups of poszare: {41}, S3, ku.