Question

In: Advanced Math

Abstract Algebra Let n ≥ 2. Show that Sn is generated by each of the following...

Abstract Algebra

Let n ≥ 2. Show that S_n is generated by each of the following sets.

(a) S₁ = {(1, 2), (1, 2, 3), (1, 2, 3, 4), ..., (1, 2, 3,..., n)}

(b) S₂ = {(1, 2, 3, ..., n-1), (1, 2, 3, ..., n)}

Expert Solution

We know that is generated by the subset of all transpositions in it.

a) Now, observe that Thus, if is the subgroup of generated by then we have

But then, for every integers . That is, contains all the transpositions in . Hence, .

b) Now, observe that . Thus, if is the subgroup of generated by then we have

But then, for every integers , we have

for every integers ; this is because is a transposition as conjugation preserves cycle structure, and sends to . Thus, contains all the transpositions in . Hence, .

Colby Messinger answered 3 years ago

Let sn = 21/n+ n sin(nπ/2), n ∈ N. (a) List all subsequential limits of (sn)....

Let sn = 21/n+ n sin(nπ/2), n ∈ N. (a) List all subsequential limits of (sn). (b) Give a formula for nk such that (snk) is an unbounded increasing subsequence of (sn). (c) Give a formula for nk such that (snk) is a convergent subsequence of (sn).

abstract algebra Let G be a finite abelian group of order n Prove that if d...

abstract algebra Let G be a finite abelian group of order n Prove that if d is a positive divisor of n, then G has a subgroup of order d.

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)>

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all...

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim sn ≥ a. (b) Show that if sn ≤ b for all but finitely many n, then lim sn ≤ b. (c) Conclude that if all but finitely many sn belong to [a,b], then lim sn belongs to [a, b].

Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m >...

Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m > 1, m ∈ N such that |sn − m| = 1/3 (this says that every term of the sequence is an integer plus or minus 1/3 ). Show that the sequence sn is eventually constant, i.e. after a point all terms of the sequence are the same

Let ? be a ?−algebra in ? and ?:? ⟶[0,∞] a measure on ?. Show that...

Let ? be a ?−algebra in ? and ?:? ⟶[0,∞] a measure on ?. Show that ?(?∪?) = ?(?)+?(?)−?(?∩?) where ?,? ∈?.

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn...

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn where k ≤ n. (a) For σ ∈ Sn, prove that στσ-1 = (σ(1), σ(2), . . . , σ(k)). (b) Let ρ be any cycle of length k in Sn. Prove that there exists an element σ ∈ Sn so that στσ-1 = ρ.

Abstract Algebra Let G be a discrete group of isometries of R2. Prove there is a...

Abstract Algebra Let G be a discrete group of isometries of R2. Prove there is a point p ∈ R2 whose stabilizer is trivial.

(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈ F[x]. Show that the...

(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈ F[x]. Show that the following properties hold: (a) If g(x)|f(x) and h(x)|g(x), then h(x)|f(x). (b) If g(x)|f(x), then g(x)h(x)|f(x)h(x). (c) If h(x)|f(x) and h(x)|g(x), then h(x)|f(x) ± g(x). (d) If g(x)|f(x) and f(x)|g(x), then f(x) = kg(x) for some k ∈ F \ {0}

Linear Algebra Carefully prove the following statement: Let A be an n×n matrix. Assume that there...

Linear Algebra Carefully prove the following statement: Let A be an n×n matrix. Assume that there exists an integer k ≥ 1 such that Ak = I . Prove that A is invertible.