Adv Alegbra HW Solutions 52

Adv Alegbra HW Solutions 52 - 52 Now | X | is even, for it...

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52 Now | X | is even, for it is the disjoint union of two-point subsets consisting of an element and its inverse. Since | G | is even, we have | Y | even. But 1 Y , and so there is at least one element g Y with g ±= 1and g 2 = 1 (indeed, there are an odd number of such elements.) 2.47 What is the largest order of an element in S n , where n = 1 , 2 ,..., 10? Solution. Denote the largest order of an element in S n by µ( n ) . There is n 1 2 3 4 5 6 7 8 9 10 µ( n ) 1 2 3 4 6 6 12 15 20 24 no known formula for µ( n ) , though its asymptotic behavior is known, by a theorem of E. Landau. 2.48 The stochastic group 6( 2 , R ) consists of all those matrices in GL ( 2 , R ) whose column sums are 1; that is, 2 , R ) consists of all the nonsingular matrices £ ac bd ± with a + b = 1 = c + d . Prove that the product of two stochastic matrices is again stochastic, and that the inverse of a stochastic matrix is stochastic. Solution. If A and B are stochastic matrices, then AB = · ¸· w y xz ¸ = · a w + cx ay + cz b w + dx by + dz ¸ . Now the sum of the entries in the f rst column of is ( a w + ) + ( b w + ) = ( a + b )w + ( c + d ) x = w + x = 1 . Similarly, the sum of the entries in the second column of
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