# LS9-17 - S UMMARY OF 9/17 L ECTURE We introduced several...

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S UMMARY OF 9/17 L ECTURE We introduced several new pieces of notation over the course of the lecture: H < G means H is a proper subgroup of G ; d | n , read “ d divides n ”, means n is a multiple of d ; and | G | , read “the order of G ”, is the number of elements of G (possibly inﬁnite). We started class with a recap of the proof that every subgroup of ( Z , +) is of the form d Z for some natural number d . Next, we turned to a curious application of the theorem. Pick any two integers a and b , and consider the set H = { ma + nb } m,n Z . This is clearly a subset of Z , but we noticed that it’s actually a subgroup. By our theorem, there exists a natural number d such that H = d Z . What’s the relationship between a,b and d ? After playing around with some examples, we ﬁgure out the following: Claim: d = gcd( a,b ) Proof: Step 1: d | a and d | b . Observe that a H . Since H = d Z , we must have a d Z , i.e. a is a multiple of d . Similarly, we conclude that

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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.

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LS9-17 - S UMMARY OF 9/17 L ECTURE We introduced several...

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