Third Isomorphism Theorem Let AT and K be normal subgroups o

Third Isomorphism Theorem). Let AT and K be normal sub-groups of G with N K. Show K/N is a normal subgroup of G/N. Define phi: G rightarrow (G/N)/(K/N) by phi (g) = (gN) (K/N) for g G. Show phi is n surjective homomorphism with ker phi = K. Conclude that (G/N)/(K/N) G/K.

Solution

Define:

: G/N G/K

by (Ng) = Kg. By the First Isomorphism Theorem, (G/N)/ ker() = (G). It is clear is onto, so (G) = G/K. Now

ker() = {Ng G/N | Kg = K} = {Ng G/N | g K} = K/N.

Thus by the First Isomorphism Theorem, (G/N)/(K/N) = G/K.