home2 - a σ A and a τ A Are the left cosets the same as...

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

View Full Document Right Arrow Icon
MATH 4107 HOMEWORK #2 DUE: February 12, 2010 Work the following problems and hand in your solutions. You may work together with other people in the class, but you must each write up your solutions independently. A subset of these will be selected for grading. Write LEGIBLY on the FRONT side of the page only, and STAPLE your pages together. 1. Let Z 8 = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } under + mod 8 , and let G = { e, r, r 2 , r 3 , a, b, c, d } be the dihedral group of order 8 that we constructed in class (the set of symmetries of the square). Do there exist homomorphisms f : Z 8 Z and g : Z G such that g f is an isomorphism? (Remember, proof is required.) 2. In the group S 3 , define permutations σ and τ as follows (I show both the cycle and the two-row notation for these permutations): σ = (1 2 3) = parenleftbigg 1 2 3 2 3 1 parenrightbigg and τ = (1 2) = parenleftbigg 1 2 3 2 1 3 parenrightbigg
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: a σ A and a τ A . Are the left cosets the same as the right cosets? 3. Suppose that f : G → H is a homomorphism, and K is a subgroup of H. We de±ne the inverse image of K under f to be f-1 ( K ) = { x ∈ G : f ( x ) ∈ K } . Show that f-1 ( K ) is a subgroup of G. Note: We are not assuming that f is a bijection, and therefore f-1 does not mean “inverse function” here. Instead, f-1 ( K ) is shorthand for a particular subset of G. 4. Problem 2.4 #16. Hint: Consider the elements in G that are their own inverses, and the ones that are not. 5. Problem 2.5 #12. 6. Problem 2.5 #16. 7. Problem 2.5 #20. Hint: Think about the element mnm-1 n-1 . 1...
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