Question

In: Advanced Math

How many elements of order 2 are there in S5 and in S6? How many elements...

How many elements of order 2 are there in S5 and in S6? How many elements of order 2 are there in Sn? (abstract algebra)

Solutions

Expert Solution

Elements of order 2 in S5 could consist of 1 two-cycles or the product of 2 two-cycles. There are, 5C2 = 10 ways to create a 2-cycle. Then there are 3C2 = 3 ways to create a second 2-cycle. So, there are 10 single 2-cycles, there are 10.3/2 = 15 pairs of disjoint 2-cycles (divide by 2 since either 2-cycle could be listed first).

So, there are 10 + 15 = 25 elements of order 2 in S5.

Elements of order 2 in S6 could consist of 1, 2, or 3 two-cycles. There are 6C2 = 15 ways to create a 2-cycle. Then there are 4C2 = 6 ways to create a second 2-cycle. Only a single way remains to create
a third 2-cycle. So there are 15 single 2-cycles, there are 15 · 6/2 = 45 pairs of disjoint 2-cycles (divide by 2 since either 2-cycle could be listed first), and 15 · 6/6 = 15 triples of disjoint 2-cycles (3! = 6 ways of ordering 3 items).
Thus there are 15 + 45 + 15 = 75 elements of order 2 in S6.


Related Solutions

how many elements in S5 and A5 have order 2
how many elements in S5 and A5 have order 2
Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with an...
Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with an experiment having the probability distribution shown in the accompanying table. If A = {s1, s2} and B = {s1, s5, s6}, find the following. Outcome Probability s1 1 3 s2 1 7 s3 1 6 s4 1 6 s5 1 21 s6 1 7 (a) P(A) = P(B) = (b) P(AC) = P(BC) = (c) P(A ∩ B) = (d) P(A ∪ B) =...
Consider the three transactions T1, T2, and T3, and the schedules S5 and S6 given below....
Consider the three transactions T1, T2, and T3, and the schedules S5 and S6 given below. Show all conflicts and draw the serializability (precedence) graphs for S5 and S6, and state whether each schedule is serializable or not. If a schedule is serializable, write down the equivalent serial schedule(s). T1: r1(x); r1(z); w1(x); T2: r2(z); r2(y); w2(z); w2(y); T3: r3(x); r3(y); w3(y); S5: r1(X); r2(Z); r1(Z); r3(X); r3(Y); w1(X); c1; w3(Y); c3; r2(Y); w2(Z); w2(Y); c2; S6: r1(X); r2(Z); r1(Z);...
2. (a) Let p be a prime. Determine the number of elements of order p in...
2. (a) Let p be a prime. Determine the number of elements of order p in Zp^2 ⊕ Zp^2 . (b) Determine the number of subgroups of of Zp^2 ⊕ Zp^2 which are isomorphic to Zp^2 .
Loop: sll $t1, $s3, 2                         Add $t1, $t1, $s6             &n
Loop: sll $t1, $s3, 2                         Add $t1, $t1, $s6                         Lw $t0, 0($t1)                         Bne $t0, $s5, Exit                         Addi $s3, $s3, 1                         j           Loop How is this MIPS instruction set translated to machine code with the loop starting at 80000 in memory?
How many subgroups of order 9 and 49 may there be in a Group of order...
How many subgroups of order 9 and 49 may there be in a Group of order 441
DISCRETE MATH If x has t elements and y has s elements, how many different one...
DISCRETE MATH If x has t elements and y has s elements, how many different one to one and onto functions are there, and what are they? Show your work
The elements of the list are going to be strings, and will be in order of...
The elements of the list are going to be strings, and will be in order of strictly increasing or decreasing length. Some examples of lists in strictly increasing length order are [“pan”, “banana”,”xylophone”], and [“kayn”, “swain”, “morgana”].   Function Name: fix_string_list Parameter: data - A list with 3 elements (that are all lowercase strings) that is either in order of strictly increasing or decreasing length. Return: A list with the same 3 elements as the parameter that is in order of...
5. (a) Let σ = (1 2 3 4 5 6) in S6. Show that G...
5. (a) Let σ = (1 2 3 4 5 6) in S6. Show that G = {ε, σ, σ^2, σ^3, σ^4, σ^5} is a group using the operation of S6. Is G abelian? How many elements τ of G satisfy τ^2 = ε? τ^3 = ε? ε is the identity permutation. (b) Show that (1 2) is not a product of 3-cycles. Must be written as a proof! (c) If a^4 = 1 and ab = b(a^2) in a...
a) Let σ = (1 2 3 4 5 6) ∈ S6, find the cycle decomposition...
a) Let σ = (1 2 3 4 5 6) ∈ S6, find the cycle decomposition of σ i for i = 1, 2, . . . , 6. (b) Let σ1, . . . , σm ∈ Sn be disjoint cycles. For 1 ≤ i ≤ m, let ki be the length of σi . Determine o(σ1σ2 · · · σm)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT