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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.)

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