hw6 - ⊆ S 4 and let K = σ ∈ S 4 | σ(4 = 4(a Show that...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 113 Homework 6, due 3/8/2012 at the beginning of section Policy: if you worked with other people on this assignment, write their names on the front of your homework. Remember that you must write up your solutions independently. 1. Fraleigh section 13, exercises 4, 5, 8, 10, 19, 49. 2. Fraleigh section 13, exercises 32, 44. 3. Fraleigh section 14, exercises 31, 34, 37. (Recall that an automorphism of G is an isomorphism from G to itself. For each a G there is an automorphism i a defined by i a ( g ) = aga - 1 ; an automorphism of G is called inner if it is i a for some a G .) 4. Let H = { id, (1 2)(3 4) , (1 3)(2 4) , (1 4)(2 3)
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: } ⊆ S 4 , and let K = { σ ∈ S 4 | σ (4) = 4 } . (a) Show that H is a subgroup of S 4 . Is H is a normal subgroup? What other group is H isomorphic to? Same questions for K . (b) Show that every coset of H contains exactly one element of K . Also show that every element of S 4 can be written uniquely as the product of an element of H and an element of K . (c) What can you say about the quotient group S 4 /H ? Is S 4 isomorphic to the direct product H × K ? 5. How challenging did you ±nd this assignment? How long did it take?...
View Full 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