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1. The problem statement, all variables and given/known data

My book states the following without any justification right before proving the Third Isomorphism Theorem: "If H and K are two normal subgroups of G and [itex]K \leq H[/itex], then H/K is a normal subgroup of G/K."

The elements of H/K are cosets of K in H. The elements of G/K are cosets of K in G. Therefore I think that statement is simply absurd. That is, the elements of H/K are not even contained in the quotient group G/K; therefore, they cannot possibly form a normal subgroup in G/K.

EDIT: wait, never mind, the cosets of K in H are also cosets of K in G; sorry

EDIT: and the reason H/K is normal in G/K is that gK*hK*g^(-1)K = (ghg^{-1})K = h' K since H is normal in G. Very EDIT: cool.

2. Relevant equations

3. The attempt at a solution

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# Normal subgroups

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