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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern