1. The problem statement, all variables and given/known data Let [itex]G[/itex] be a group with a normal subgroup [itex]N[/itex] and subgroups [itex] K \triangleleft H \leq G. [/itex] If [itex]H/K [/itex] is nontrivial, prove that at least one of [itex]HN/KN[/itex] and [itex](H\cap N)/(K\cap N)[/itex] must be nontrivial. 2. Relevant equations The Three (or Four) Isomorphism Theorems. 3. The attempt at a solution By the first isomorphism theorem, we saw that [itex]HN/KN \cong H/K [/itex]. So if [itex]H/K[/itex] is nontrivial, then [itex]HN/KN [/itex] is also nontrivial. Now to show that [itex](H\cap N)/(K\cap N)[/itex] is also nontrivial, what normal subgroup of [itex]H/K [/itex] is this quotient group [itex](H\cap N)/(K\cap N)[/itex] isomorphic to? Because of the "and" in the statement of the problem, should both be nontrivial?