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## Main Question or Discussion Point

If G is a group with subgroups H and K, and if H is normal, then HK=KH, right?...Since we know that N(H) = G. However, since H commutes with K, then H must also be contained in N(K), right?...Since N(K) is the set of elements that commute with K...but I was a bit confused, because we know that, for any subgroup M, GM=MG. So G must be contained in the normalizer of any subgroup, right? But that doesn't make sense, since the normalizer of any subgroup would then need to equal G and every subgroup would be normal...I know I'm probably missing something, but I'm not sure what. So I was wondering if anybody could clarify this for me.

Thanks in advance

Thanks in advance