Thx a lot!
I am a little confused about the reason {N,G/N} = {M,G/M} implying that G/M=N, and G/N=M , why we cannot get G/N isomorphic to G/M, M isomorphic to N? And I have a stupid question, the intersection of N and G/N would be always trivial or not?
If we can get G= M(semi-direct product)N and G =N(semidirect product)M,then since both M and N are normal subgroups, so the semi-direct products are direct,that's due to the definition.
And I am also thinking of using the facts: M N are bothe maximal normal subgroups, (We can gurantee here that M and N are both maximal, but whether the intersection is trivial? ) then G = MN