(a) Prove that Sn is generated by the elements in the set {(i
i+1) : 1≤i≤n}.
[Hint: Consider conjugates, for example (2 3) (1 2) (2
3)−1.]
(b) ProvethatSn isgeneratedbythetwoelements(12)and(123...n) for
n ≥ 3.
(c) Prove that H = 〈(1 3), (1 2 3 4)〉 is a proper subgroup of
S4.
Sn = (1+(1/n))^n
(a) Prove Sn is strictly increasing (b) bounded below by 2 and
above by 3
(c) Sn converges to e
(d) Obtain an expression for e
(e) Prove e is irrational
Abstract Algebra
Let n ≥ 2. Show that Sn 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)}
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 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).
Prove combinatorially: {{n \ choose x)= {{k+1 \ choose n-1)
This question is asking to prove combinatorially
whether n multi-choose k is equal to k+1 multi-choose
n-1
Thank you