The most difficult part of this material for me was understanding the symmetric group on n symbols. I am not sure how they came to all of these generalizations about Sn by just having looked at S6. I also found it confusing to try and see how they came up with the inverse of f under composition. I also don't understand how S3 had order 6???? I thought if elements repeated then they were just counted as 1 element within a set. Is this just a process of trial and error or is there some method to this? I was also wondering exactly what a group is, is it is "larger" than a ring, and if all groups are comparable to permutations.
The most interesting part of this material for me was the properties of S3 under composition. In another one of my math classes we are showing that if f and g are continuous, then so is their composition. I was thinking about this and trying to relate it to the example in this section. I am not sure how it relates, but it made me wonder if being continuous had anything to do with the fact that the composition operation is a group when dealing with integers.
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