QuestionQuestion

Transcribed TextTranscribed Text

3. (Exploring Cycle products). Consider the symmetric group Sn. Consider products of the form aod where a is a 2-cycle and o is a 4-cycle. a) Consider the three cases a no o where (i) shares elements with (e.g. (ab) (cdef)(ab)), (ii) where out a shares one element with o (e.g. where (ad) (bcde) (ad)) and (ii) both elements of a are in o. Try several examples. Describe the effect of a on the 4-cycle in each case. b) Test out your conjecture a case o a o a for where is 5-cycle. Also where is disjoint 3-cycle and 2-cycle. Does it work when a is a 3-cycle? c) Now assume a # B are two distinct 2-cycle and let o be a 3-cycle. Find a product of the form ao B that is a 5 cycle. Thus the product of elements of orders 2 and 3 can give an element of order 5. Can this ever happen in an abelian group? 4. (Elements of order 2). Let G be a group of even order, i.e., IGI is even. a) Show that the number of elements of order 2 is always odd. Hence every group of even order has at least one element with order 2. (Hint: Elements of other orders come in pairs of a and a-¹.) For any odd number n find a non-abelian group with exactly n elements of order 2. b) Now let G be an abelian group. We have seen if a # b and lal = 161 = 2 then K = {e, a, b, ab} is a subgroup with three distinct elements of order 2. Show if d € G is another different element with |d| = 2 then G will have at least 7 distinct elements of order 2. Hence abelian groups cannot have 5 elements of order 2. (Hint: Consider dK = {dk I k € K}. Show these are all distinct order 2.) c) Show that if a E G is the only element of order 2 in G then a is in the center of the group. That is, aE Z(G). (Hint: Suppose a & Z(G). Use ab # ba to create another distinct element of order 2.) d) Let H < G be any subgroup and let a € Z(G) be of order 2. Show H U aH is a subgroup of G. (Note: Recall aH = {ah I H}. 5. (The Order of a product ab). a) Let a be an element of a group G with order |a| = n. Prove that if ak = e then n must be a divisor of k. (Hint: use the division algorithm to write k = nq + r and proceed to show that r must be 0.) b) In abelian groups it is often true that ab is the least common multiple of lal and 161. But this does not always hold! Suppose a is an element of order 12 and b = a2. What is the order of labl and how does it relate to the orders of a and b alone? Prove that the order of labl must be a divisor of the least common multiple. c) Prove that in an abelian group G if (b) n (a) = e then the order (ab) is the least common multiple of the orders |a| = n and 161 = m. (Hint: Suppose (ab)k = e, show that (ak) C (b) and (bk) C (a). Use part (a) to conclude that k must be a multiple of both m and n.) d) We now drop the assumption that G is necessarily abelian. Prove in any group it is always true that Jab| = lbal. (Hint: Let (ab) have order n and consider the product b(ab)" and move parentheses.)

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:
    Solution.zip.

    $40.00
    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 Abstract Algebra 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.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    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