Transcribed TextTranscribed Text

Problem 3. Let G be a group. a) Let [IOTA] e(x) = x-1 for all x in G. i. Show that , € Sym(G). ii. Suppose G has order n and let a = ca(2). What is the cycle type of c? iii. Deduce that if G has even order then G has an element of order 2. b) Fix g E G and define [RHO] p(x) = g.r for all x in G. i. Show that P E Sym(G). ii. Suppose G has order n and let b = o(g). What is the cycle type of p? iii. Deduce that o(g) I o(G) without appealing to Lagrange's Theorem. Problem 4. Fill in the following table for S6 like we did for Sa in Lecture 11. partition cycle type order number : : : : Using the table, determine (d) for all d. What is exp(S6)? Problem 5. Find explicit elements of S = Sym(N) with the following cycle types: a) (1,1,1, ;0) b) 1) c) (1,1,1, 1) d) (bonus) (0,0,0, 00). [Your answers must be unambiguous. That does not mean you need to come up with formulas or piecewise definitions; you just need to make it clear what the output would be given any positive integer input. For example, you can write out a small part of each permutation (using two-line or cycle notation) and indicate the pattern or the rest of the definition in words. No proofs required. The bonus must be flawless to count as correct.]

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Mathematics - Other Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats