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Kernel and Homomorphism

Here's my problem:

If A and B are subsets of a group G, define
AB = {ab | a 2 A, b 2 B}. Now suppose phi: G -> G0 is a homomorphism of groups and N = Ker(phi) is its kernel.

(i) If H is a subgroup of G, show that HN = NH. (Warning: this is an equation of sets; proceed
accordingly; do not assume that G is abelian.)

(ii) Show that phi-inverse[phi[H]] = HN.


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Solution Summary

Kernels and Homomorphisms are investigated. The solution is detailed and well presented.