Paper 3, Section II, D

Groups | Part IA, 2019

State and prove the first isomorphism theorem. [You may assume that kernels of homomorphisms are normal subgroups and images are subgroups.]

Let GG be a group with subgroup HH and normal subgroup NN. Prove that NH={nh:nN,hH}N H=\{n h: n \in N, h \in H\} is a subgroup of GG and NHN \cap H is a normal subgroup of HH. Further, show that NN is a normal subgroup of NHN H.

Prove that HNH\frac{H}{N \cap H} is isomorphic to NHN\frac{N H}{N}.

If KK and HH are both normal subgroups of GG must KHK H be a normal subgroup of GG ?

If KK and HH are subgroups of GG, but not normal subgroups, must KHK H be a subgroup of GG ?

Justify your answers.

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