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Visual Group Theory, Lecture 3.6: Normalizers
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Visual Group Theory, Lecture 3.6: Normalizers
A subgroup H of G is normal if xH=Hx for all x in G. If H is not normal, then the normalizer is the set of elements for which xH=Hx. Obviously, the normalizer has to be at least H and at most G, and so in some sense, this is measuring "how close H is to being normal". We interpret this in terms of Cayley diagrams, and then prove some basic properties of normalizers: they are always subgroups, and they are unions of cosets -- precisely, those left cosets that are also right cosets.
A subgroup H of G is normal if xH=Hx for all x in G. If H is not normal, then the normalizer is the set of elements for which xH=Hx. Obviously, the normalizer has to be at least H and at most G, and so in some sense, this is measuring "how close H is to being normal". We interpret this in terms of Cayley diagrams, and then prove some basic properties of normalizers: they are always subgroups, and they are unions of cosets -- precisely, those left cosets that are also right cosets.
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