Math 103A Winter 2001 Professor John J Wavrik Normal Subgroups Take a look at the list of subgroups of various groups generated by the Groups32 SUBGROUPS command G1 SUBGROUPS wait of Group Number 7 Normal subgroup Generators 0 1 D 2 C 3 B Subgroup A A D A C E A B C D E F G7 SUBGROUPS wait of Group Number 8 Normal subgroup Generators 0 1 D 2 E 3 F 4 B 5 B D Subgroup A A D A E A F A B C A B C D E F G8 CHART Order of Groups 1 32 or 0 Number 6 7 8 There are 2 Groups of order 6 1 abelian and 1 non abelian Notice that some of these are marked as normal There are several equivalent ways to define what it means for a subgroup to be normal Definition A subgroup H of a group G is called normal if any one of the following conditions holds 1 2 3 4 5 g G h H we have ghg 1 H g G gHg 1 H g G gH Hg Every right coset of H is a left coset H is the kernel of a homomorphism of G to some other group It is easy to see from condition 1 that Proposition If G is an abelian group then any subgroup is normal Group 7 above is an abelian group all of its subgroups are normal In group 8 above the subgroup F is not normal while the subgroup B is Lets check the cosets of these two subgroups G8 COSETS of subg generated by set f Left Cosets A F B E C D Right Cosets A F B D C E The subgroup A F G8 COSETS is NOT a NORMAL subgroup of subg generated by set b Left Cosets A B C D E F The subgroup A B C Right Cosets A B C D E F is a NORMAL subgroup 6 If H F we see that the right coset HB B D does not coincide with any of the left cosets H is not a normal subgroup for this reason We also can see that condition 1 is not satisfied We must find a g G and h H so that ghg 1 lies outside H G8 EVALUATE use for inverse bfb D We can also see this using permutations Group 8 is isomorphic to S3 The elements B and C are of order 3 so must be 3 cycles The elements D E F are of order 2 so are 2 cycles Let H be the subgroup 1 2 Let g 1 2 3 1 2 3 1 2 1 2 3 1 2 3 which is not in H Please note to show H is not normal we only have to find one g and one h for which ghg 1 H The earlier conditions are usually the easiest to verify The importance of normal subgroups however come from the fact that they are kernels of homomorphisms and that the set of cosets can be made into a group Let G be a group and H a subgroup The cosets are the equivalence classes for congruence mod H We would like to imitate what was done for integers mod n and define a b a b to make the equivalence classes a group We will see that this only works when H is normal