Show that {t_(1,s) : 2 ≤ s ≤ n} is a minimal generating set for S_n....

Show that {t_(1,s) : 2 ≤ s ≤ n} is a minimal generating set for S_n. You may use the fact that {t_(r,s) : 1 ≤ r < s ≤ n}, as defined in the outline, generates S_n.

Expert Solution

It is required to show that the (n-1) transpositions as above is a minimal generating set for Sn. The proof submitted involves 3 parts. First it is proven that transpositions generate Sn. Then it is proven that the given (n-1) transpositions generate Sn. Finally it is shown that the given (n-1) transpositions is a minimal generating set for Sn.

Colby Messinger answered 2 years ago

Find the minimal generating set of G -- G = Z2 × Z3 -- G =...

Find the minimal generating set of G -- G = Z2 × Z3 -- G = Z2 × Z2 -- G = Z × Z

a. What are the maximal and minimal elements, if any, of the set (N+,|)? Is there...

a. What are the maximal and minimal elements, if any, of the set (N+,|)? Is there a minimum or maximum element? (N+={1,2,3,4,...}). b. There are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla. We can have three scoops. How many variations will there be?

Q1- Given a set S = {1, 2, . . . , n} of players, with...

Q1- Given a set S = {1, 2, . . . , n} of players, with skill levels a1, a2, . . . , an, we need to partition the set of players S into two teams S1 and S2 of equal total skill. The teams do not need to be of equal size. However, every player must be in exactly one team. In other words, S1 ∪ S2 = S (1) S1 ∩ S2 = ∅ (2) Σ ak=...

(1) Show that the set { 1 m + 1 n : m, n ∈ N}...

(1) Show that the set { 1 m + 1 n : m, n ∈ N} is countable. (2) Show that the set {a + b √ 2 : a, b ∈ Q} is countable. (3) Show that the intersection of two countable sets is countable. (4) Show that the set of all irrational numbers is uncountable. (5) Let C = {0, 1, 2, . . . , 9}. Show that the set C ×C × · · · is...

For any r, s ∈ N, show how to order the numbers 1, 2, . ....

For any r, s ∈ N, show how to order the numbers 1, 2, . . . , rs so that the resulting sequence has no increasing subsequence of length > r and no decreasing subsequence of length > s.

1)Show that a subset of a countable set is also countable. 2) Let P(n) be the...

1)Show that a subset of a countable set is also countable. 2) Let P(n) be the statement that 13 + 23 +· · ·+n3 =(n(n + 1)/2)2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive step, identifying where you use the inductive hypothesis....

(a) Show that the sample variance s 2 = [Pn i=1(xi − x¯) 2 ]/(n −...

(a) Show that the sample variance s 2 = [Pn i=1(xi − x¯) 2 ]/(n − 1) can also be expressed as s 2 = [Pn i=1 x 2 i − ( Pn i=1 xi) 2 n ]/(n − 1). At a medical center, a sample of 36 days showed the following number of cardiograms done each day. 25 31 20 32 20 24 43 22 57 23 35 22 43 26 56 21 19 29 36 32 33 32...

Show that (1 + 2 +. . .+n)2 > 12 +. . .+ n2, for n...

Show that (1 + 2 +. . .+n)2 > 12 +. . .+ n2, for n ≥ 2.

Show that (a)Sn=<(1 2),(1 3),……(1 n)>. (b)Sn=<(1 2),(2 3),……(n-1 n)> (c)Sn=<(1 2),(1 2 …… n-1 n)>

1: Let X be the set of all ordered triples of 0’s and 1’s. Show that...

1: Let X be the set of all ordered triples of 0’s and 1’s. Show that X consists of 8 elements and that a metric d on X can be defined by ∀x,y ∈ X: d(x,y) := Number of places where x and y have different entries. 2: Show that the non-negativity of a metric can be deduced from only Axioms (M2), (M3), and (M4). 3: Let (X,d) be a metric space. Show that another metric D on X can...

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