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4.2 cor 5o - [16.1 Let p be the smallest prime dividing the...

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( January 14, 2009 ) [16.1] Let p be the smallest prime dividing the order of a finite group G . Show that a subgroup H of G of index p is necessarily normal . Let G act on cosets gH of H by left multiplication. This gives a homomorphism f of G to the group of permutations of [ G : H ] = p things. The kernel ker f certainly lies inside H , since gH = H only for g H . Thus, p | [ G : ker f ]. On the other hand, | f ( G ) | = [ G : ker f ] = | G | / | ker f | and | f ( G ) | divides the order p ! of the symmetric group on p things, by Lagrange. But p is the smallest prime dividing | G | , so f ( G ) can only have order 1 or p . Since p divides the order of f ( G ) and | f ( G ) | divides p , we have equality. That is, H is the kernel of f . Every kernel is normal, so H is normal. /// [16.2] Let T Hom k ( V ) for a finite-dimensional k -vectorspace V , with k a field. Let W be a T -stable subspace. Prove that the minimal polynomial of T on W is a divisor of the minimal polynomial of T on V . Define a natural action of T on the quotient V/W , and prove that the minimal polynomial of T on V/W is a divisor of the minimal polynomial of T on V . Let f ( x ) be the minimal polynomial of T on V , and g ( x ) the minimal polynomial of T on W . (We need the T -stability of W for this to make sense at all.) Since f ( T ) = 0 on V , and since the restriction map End k ( V ) End k ( W ) is a ring homomorphism, (restriction of) f ( t ) = f (restriction of T ) Thus, f ( T ) = 0 on W . That is, by definition of g ( x ) and the PID-ness of k [ x ], f ( x ) is a multiple of g ( x ), as desired. Define T ( v + W ) = Tv + W . Since TW W , this is well-defined. Note that we cannot assert, and do not need, an equality TW = W , but only containment. Let h ( x ) be the minimal polynomial of T (on V/W ). Any polynomial p ( T ) stabilizes W , so gives a well-defined map p ( T ) on V/W . Further, since the natural map End k ( V ) End k ( V/W ) is a ring homomorphism, we have p ( T )( v + W ) = p ( T )( v ) + W = p ( T )( v + W ) + W = p ( T )( v + W ) Since f ( T ) = 0 on V , f ( T ) = 0. By definition of minimal polynomial, h ( x ) | f ( x ). /// [16.3] Let T Hom k ( V ) for a finite-dimensional k -vectorspace V , with k a field. Suppose that T is diagonalizable on V . Let W be a T -stable subspace of V . Show that T is diagonalizable on W . Since T is diagonalizable, its minimal polynomial f ( x ) on V factors into linear factors in k [ x ] (with zeros exactly the eigenvalues), and no factor is repeated. By the previous example, the minimal polynomial g ( x ) of T on W divides f ( x ), so (by unique factorization in k [ x ]) factors into linear factors without repeats. And this implies that T is diagonalizable when restricted to W . /// [16.4] Let T Hom k ( V ) for a finite-dimensional k -vectorspace V , with k a field. Suppose that T is diagonalizable on V , with distinct eigenvalues . Let S Hom k ( V ) commute with T , in the natural sense that ST = TS . Show that S is diagonalizable on V .
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