52Now|X|is even, for it is the disjoint union of two-point subsets consistingof an element and its inverse. Since|G|is even, we have|Y|even. But1∈Y, and so there is at least one elementg∈Ywithg=1 andg2=1(indeed, there are an odd number of such elements.)2.47What is the largest order of an element inSn, wheren=1,2, . . . ,10?Solution.Denote the largest order of an element inSnbyµ(n). There isn12345678910µ(n)12346612152024no known formula forµ(n), though its asymptotic behavior is known, by atheorem of E. Landau.2.48Thestochastic group(2,R)consists of all those matrices in GL(2,R)whose column sums are 1; that is,(2,R)consists of all the nonsingularmatricesa cb dwitha+b=1=c+d.Prove that the product of two stochastic matrices is again stochastic, andthat the inverse of a stochastic matrix is stochastic.Solution.IfAandBare stochastic matrices, thenAB=acbdwyxz=aw+cxay+czbw+dxby+dz.Now the sum of the entries in thefirst column ofABis(aw+cx)+(bw+dx)=(a+b)w+(c+d)x=w+x=1.Similarly, the sum of the entries in the second column of
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