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Unformatted text preview: S UMMARY OF 9/24 L ECTURE We started lecture by talking more about Z n , a cyclic group of order n which is generated by the element 1 + subject to the condition n 0 ( mod n ) . In other words, Z n = { , 1 , 2 ,...,n 1 } under the binary operation addition ( mod n ) : just like ordinary addition except that n is the same thing as 0, which we write in the form n 0 ( mod n ) . From this it follows that n + 1 1 ( mod n ) , n + 5 5 ( mod n ) , etc. A natural question is, if we can do addition ( mod n ) on Z n , how about multiplication ( mod n ) ? In other words, is Z n a group under multiplication ( mod n ) ? Well, its certainly closed, asso ciative, and has an identity namely, 1. But, not every element is an inverse... as usual, 0 isnt. (Because 0 times anything is 0, and therefore will never give the identity.) OK, so we remove 0 from Z n . Now do we have a group under multiplication? After some playing around, we determined that the answer is, sometimes. The way we played around withplaying around, we determined that the answer is, sometimes....
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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 Toronto.
 Fall '10
 GideonMaschler
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