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Let H be a normal subgroup of G and K a subgroup of G that contains
H. Prove that K is normal in G if and only if K/H is normal in G/H.
Assume that H is a normal subgroup of G, and that K is a subgroup of G containing H. Then, we need to show that K is normal in G if and only if K/H is normal in G/H.
First, suppose that K is normal in G. We need to show that K/H is normal in G/H. Let gH be an arbitrary element of G/H, where g is an element of G. Then, we need to show that (gH)(K/H)(gH)^(-1) is a subset of K/H. Let kH be an arbitrary element of K/H. Then, we have
(gH)(kH)(gH)^(-1) = (gkg^(-1))H,
where k is an element of K. Since K is normal in G, we have gkg^(-1) is an element of K. Thus, we have (gH)(kH)(gH)^(-1) = (gkg^(-1))H is an element of K/H. Therefore, K/H is normal in G/H.
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