LS9-22 - S UMMARY OF 9/22 L ECTURE We started class by...

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S UMMARY OF 9/22 L ECTURE We started class by considering the following matrix: A := ± 0 1 - 1 0 ² We noted that A 2 = - I , that A 4 = I , and that A SL 2 ( Q ) (which means that A is also in GL 2 ( Q ) ). Some of you, in your work on problem 1.5, made the false claim that if g n = I for some matrix g , then g must be the identity; the matrix A above provides a counterexample. I also pointed out that I’d made an error in my proof (given during the 9/10 lecture) that the group Q × is not cyclic: if α were a generator of Q × and α n = 1 , we can only deduce that α = ± 1 . (In class I claimed that α would have to equal 1 under the circumstances.) Of course, the proof can still be made to work, since neither 1 nor - 1 generates Q × . Having defined the matrix A , we examined the cyclic group generated by it: { 1 ,A,A 2 ,A 3 } ≤ SL 2 ( Q ) . This is highly reminiscent of some subgroups we’ve run across before, namely { 1 ,i,i 2 ,i 3 } ≤ C × and { 1 ,α,α 2 3 } ≤ { symmetries of the square
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LS9-22 - S UMMARY OF 9/22 L ECTURE We started class by...

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