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Unformatted text preview: S UMMARY OF 10/8 L ECTURE We started by clearing up a couple of possible points of confusion about the nature of sets: { 1 , 2 , 2 , 3 } is a set with 3 elements, not 4. Also, { 1 , 2 , 3 } = { 2 , 3 , 1 } – the set is just a bag containing a bunch of elements, in no particular order. We next returned to the material from the end of the previous lecture. Using the “sudoku principle” we wrote down all possible multiplication tables for some groups of small order. We found that there is only one possible group of order 2 – the cyclic group. Similarly, we found that the only group of order 3 is the cyclic group. For groups of order 4, the picture changes: this time there were four possible multiplication tables. Of these, three were the same (up to relabeling of the elements): they were just the cyclic group of order 4. The other possibility is the group { 1 ,a,b,c } satisfying a 2 = b 2 = c 2 = 1 (this completely determines the multiplication table). This group is called the Klein V group. We next moved on to consider groups of order 5. This time, rather than studying multiplication tables, we used Lagrange’s theorem. Given G of order 5, pick any element a ∈ G and use it to generate a cyclic subgroup of...
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 Fall '10
 GideonMaschler
 Sets, Isomorphism, Cyclic group, Klein V group, possible multiplication tables

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