4.2 cor 5 +

# 4.2 cor 5 + - 7.3.12 If G is a nite group of order n and p...

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7.3.12 . If G is a °nite group of order n , and p is the least prime such that p j n , show that any subgroup of index p is normal in G . Let H be a subgroup of index p in G . Consider the action of G on the set S of left cosets of H given by a ° xH = axH as in Exercise 7.3.2. Let be the corresponding homomorphism from G to Sym S , that is, ( a ) ( xH ) = axH . Note that ker ± H because if a 2 ker , then aH = H . Now G= ker is isomorphic to a subgroup of Sym S , by the fundamental homomorphism theorem, so j G= ker j divides j Sym S j = p ! . Also j G= ker j divides j G j = n . As p is the smallest prime divisor of n , it follows that j G= ker j is either 1 or p . In the °rst case, ker = G , which is impossible because ker ± H . So j G= ker j = p , the index of H in G . But ker ± H , so ker = H , whence H is normal because it is the kernel of a homomorphism. 7.4.1 . Let G be a °nite abelian group, and let p be a prime divisor of j G j . Show that the Sylow p - subgroup of G consists of e and all elements whose order is a power of p .

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