Question

Let n ≥ 2. Show that exactly half of the permutations in Sn are even , by finding a bijection from the set of all even permutations in Sn to the set of all odd permutations in Sn.

Answer #1

Suppose that the set An has all the even permutations
in n-permutations. Prove that this set is the same as [a set
consisting of cyclic permutations of length 3 and their
products].

Define sequences (sn) and (tn) as follows: if n is even, sn=n
and tn=1/n if n is odd, sn=1/n and tn=n, Prove that both (sn) and
(tn) have convergent subsequences, but that (sn+tn) does not

Let H ={σ∈Sn |σ(n) = n}. Show that H ≤ Sn
and H∼= Sn-1.

Let s1 := 1 and
Sn+1 := 1 + 1/sN n element N
Show that (Sn) has limit L and that l can be
explicitly computed. What is the limit?

(a) Let N be an even integer, prove that GCD (N + 2, N) = 2.
(b) What’s the GCD (N + 2, N) if N is an odd integer?

If kr<=n, where 1<r<=n. Prove that the number of
permutations α ϵ Sn, where α is a product of k disjoint r-cycles is
(1/k!)(1/r^k)[n(n-1)(n-2)...(n-kr+1)]

Let S_n be the collection of permutations on {1,2,...,n}.
Consider the cycle s=(1,2,...,n) and consider the cyclic group
generate by s, denoted <s>. Prove that the set all t in S_n
such that ts=st, is just the set <s>

Show that Sn = <(1, 2), (1, 2 , ..., n)>

Show that if G is a subgroup of Sn and |G| is odd, then G is a
subgroup of An, given n≥2

If sn = 1+ 1/2 + 1/3 + 1/4 +···+ 1/n show that S 2^n ≥ 1+n/2 for
all n.
Elementary Real Analysis

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