512.42LetG=GL(2,Q), and letA=£0−110±andB=£01−11±. Show thatA4=I=B6, but that(AB)n±=Ifor alln>0. Conclude thatcanhave infnite order even though both factorsAandBhavefnite order.Solution.=·¸and()n=·1n¸.2.43(i)Prove, by induction onk≥1, that·cosθ−sinθsinθcosθ¸k=·coskθ−sinkθsinkθcoskθ¸.Solution.The proof is by induction onk. The base step is obvious.For the inductive step, letA=·cosθ−sinθsinθcosθ¸.ThenAk+1=AAk, and matrix multiplication gives the desiredresult if one uses the addition formulas for sine and cosine.(ii)Find all the elements offnite order inSO(2,R), the special or-thogonal group.Solution.By part (i), a matrixA=·cosθ−sinθsinθcosθ¸.hasfnite order if and only if coskα=1 and sinkα=0; that is,whenkαis an integral multiple of 2π. Thus,Ahasfnite order ifα=2π/kfor some nonzero integerk.2.44IfGis a group in whichx2=1 for everyx∈G, prove thatGmust be
This is the end of the preview. Sign up
access the rest of the document.
This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.