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Unformatted text preview: S UMMARY OF 9/8 L ECTURE Let G be a set (of numbers, or names, or countries, or points in the plane, or fruits, or whatever). A binary operation on G is a rule for how to combine any two elements of G to get a third. More precisely, @ is a binary operation on G if for any two elements a,b ∈ G , a @ b is an element of G . We discussed a bunch of examples of sets with binary operations. Some of these were partic- ularly nice for one reason or another; after playing around with examples, we discovered all the ‘particularly nice’ examples all had two properties in common, which were absent from the not-so-nice examples: (a) given any elements g,h,k ∈ G , g @ h @ k is unambiguous; and (b) given any two elements g,h ∈ G , one can "get" from g to h . We’ll discuss these two properties more formally after writing down the examples we con- sidered in lecture. In the meantime, any set G with a binary operation satisfying these two properties we’ll call a group . One last note before getting to the examples; we noticed that to check that (b) holds, one usually identifies a special element of G which is easy to get to, and from which it’s easy to get to any other element of the set. In each of the examples below, I’ll make a note of what the special element is.special element is....
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This note was uploaded on 03/26/2011 for the course MAT 301 taught by Professor Gideonmaschler during the Fall '10 term at University of Toronto.
- Fall '10