Algebra

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

Kuperberg (10/17/07) Math 150a: Modern Algebra First Midterm Solutions 1. Show that every finite group G has an even number of elements of order 3. Solution: In general, for any element g of any group, the orders of g and g - 1 are the same. This comes from the fact that g n = e g - n = e . Now, if g G has order 3, then g - 1 also has order 3, and we can begin to count off such elements (of order 3) in pairs. There won’t be any double counting, since g = g - 1 would imply that g has order 2. Since G is a finite group, we are also guaranteed to be finished counting at some point. Thus, we will have an even number of elements of order 3. 2. Consider a product a = xyzw of four transpositions in some symmetric group S n . Can this product be a 4-cycle? No. A k -cycle must have signature ( - 1 ) k - 1 . So an even permu- tation, such as a , cannot be a cycle of even length. Can it be a 5-cycle? Yes. Here’s an example: ( 1 5 )( 1 4 )( 1 3 )( 1 2 ) = ( 1 2 3 4 5 ) . Can it be a 9-cycle? No. Each transposition affects two numbers, while every other number is sent to itself. So the product of four transpositions can, at most, act nontrivially on 8 num-
Image of page 1

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

Image of page 2
This is the end of the preview. Sign up to access the rest of the document.
  • Spring '03
  • Kuperberg
  • Algebra

{[ 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