transpositions Definition and 19 Threads

There is a proof that shows by induction (and by contradiction) that the identity permutation decomposes into an even number of transpositions. The proof is presented in the first comment here...

Question: In defining adjacent transpositions in a permutation as swaps between neighbors, is one referring to the original set or to the last result before the transposition is applied? I clarify with an example. Suppose one assumes a beginning ordered set of <1,2,3> It is clear that (1,2)...

Homework Statement Attached are some screen shots of portion of the textbook I'm currently working through: Homework EquationsThe Attempt at a Solution My first question, why exactly can't ##\Delta## contains ##x_p - x_q## only once (note, switched from ##i,j## to ##p,q##)? As you can see...

is there any easier way of proving that no matter how an identical permutation say (e) is written the number of transpositins is even. my work i tried let t_1...t_n be m transpositions then try to prove that e can be rewritten as a product of m-2transpositions. i had x be any numeral appearing...

Hi all, I've answered a question but there's no answer for it, and if ye could tell me if I'm doing it right I'd appreciate it thanks :) Permutation: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 --------------------2 3 1 6 5 4 8 10 13 11 12 7 14 9 (i) Write it as a product of transpositions. I...

Homework Statement Show that every transposition (i,j)(1≤i≤j≤n) in Sn is expressible as a product of adjacent transpositions. Also express the transposition (1,9) as a product of adjacent transpositions. Homework Equations none The Attempt at a Solution Really struggling to even...

Homework Statement Hi! There's a theorem 7.43 in p.221(Hungerford's abstract algebra) which states that every permutation in S_{n} is a product of transpositions. What I know about the concept of transposition is it is defined if there are at least two distinct elements. But, in the above...

Hi, Homework Statement Can all permutations of {A,B,C,D} be made by multiplications of transpositions (AB), (BC), (CD)? And by multiplications of transpostion (AB) and 4-cycle (ABCD)? What is the maximum number of multiplications needed in both cases? Homework Equations All...

Homework Statement Write the permutation P= 12345678 23156847  in cycle notation, and then write it as a product of transpositions Homework Equations The Attempt at a Solution I got the cycle notation to be (123)(45687), but i am now not sure now to write it as a product of...

Hi all, long time reader first time poster! Just need a hand on this problem I've been stuck on for a few days Homework Statement Let r=(a_1,a_2...a_k) be in S_n. Suppose that ß is in S_n. Show that: ßrß^-1=(ß(a_1), ß(a_2)...ß(a_k)). Homework Equations The Attempt at a...

Homework Statement (1 2 3 4 5) (2 1 3 5 4) Write the bottom number as a product of transpositions Homework Equations The Attempt at a Solution 41352 41325 41235 42135 42315 24315 24351 21354 (2 4) (2 5) (2 3) (2 1)...

Homework Statement 1)Consider the permutation in S3 = ( 1 2 3 ) ( 1 2 3 ) NOTE: the two pairs of parenthesis are meant to be one pair that encases both rows Write as a product...

Homework Statement 5.1: Prove that S_n is generated by the set {(1 2), (3 4),...,(n-1 n)}Homework Equations None that I know ofThe Attempt at a Solution Any element in S_n can be written as a product of disjoint n-cycles. So now I need to show any n-cycle can be written as a product of...

Let α (alpha) all in S_n be a cycle of length l. Prove that if α = τ_1 · · · τ_s, where τ_i are transpositions, then s geq l − 1. I'm trying to get a better understanding of how to begin proofs. I'm always a little lost when trying to solve them. I know that I want to somehow show that s is...

Homework Statement Let α (alpha) all in S_n be a cycle of length l. Prove that if α = τ_1 · · · τ_s, where τ_i are transpositions, then s geq l − 1.Homework Equations The Attempt at a Solution What I was actually looking for is where to start with this proof. I don't want the answer, just a...

Hi, Was wondering if anyone could explain to me what an adjacent transposition is (in relation to permutations, cycles etc). I know what a transposition is, eg the product of transpositions for (34785) would be (35)(38)(37)(34). I don't know what an adjacent transposition is though...

Hello there. Can someone help me understand the following practical physics transpositions. I would like every last detail to be mentioned as I'm not really very sure on this at all. A submarine has a maximum allowable pressure of 588.6kPa. It is in water at its usual density of 1000kg/m^3...

Can someone help me? I need to prove that for m>=2, m permutations can be written as at most m-1 transpositions. I can't figure this out for the life of me! thanks in advance :confused:

Hello, I am a little confused about an example. By definition, A cycle of m symbols CAN be written as a product of m - 1 transpositions. (x1 x2 x3 ... xn) = (x1 x2)(x1 x3)...(x1 xn) Now Express the permutation (23) on S = {1,2,3,4,5} as a product of transpositions. (23) =...