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Unformatted text preview: S UMMARY OF 9/10 L ECTURE Last time we discussed associativity as though it were a special property, but when does it fail? Here’s one example: what does 4 ÷ 2 ÷ 2 mean? Well, it could mean (4 ÷ 2) ÷ 2 = 1 , or it could mean 4 ÷ (2 ÷ 2) = 4 . The point is, the expression 4 ÷ 2 ÷ 2 is ambiguous; so, for example, ( Q × , ÷ ) is not a group, because associativity fails to hold. (There are other reasons it’s not a group, as well.) Next, we proved that any group has a unique identity element (hence, we talk about the identity) and that every element has a unique inverse. Of our three examples of groups so far, recall that example (v) was a bit different from the other two: f ◦ g 6 = g ◦ f in general. However, for some choices of f and g the two expressions are equal; for example, 1 commutes with every element f ∈ A ( S ) . More generally, given a group ( G, @ ) , we say two elements a,b ∈ G commute if a @ b = b @ a . If every element of G commutes with every other element, then we call...
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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