The Third Isomorphism Theorem

  1. Suppose N is a normal subgroup of a group G and Ο€N:Gβ†’G/N is the usual projection homomorphism, defined by Ο€N(g)=gN. Prove that if Ο•:Gβ†’H is any homomorphism with N≀ker⁑(Ο•), then there exists a unique homomorphism ψ:G/Nβ†’H such that Ο•=Οˆβˆ˜Ο€N. (You must explicitly define ψ, show it is well defined, show Ο•=Οˆβˆ˜Ο€N, and show that ψ is uniquely determined.)
  2. Prove the:
    Third Isomorphism Theorem. If M,N⊴G with N≀M, then (G/N)/(M/N)β‰…G/M.