Question

a. Prove for all σ, τ ∈ Sn that στσ−1 τ −1 ∈ An.

b. Let p and q be distinct odd primes. Prove that
Z^{x}_{pq} is not a cyclic group.

Answer #1

Let σ ∈ Sn.
a) Prove that σ is even if and only if σ−1 is even.
b) Prove that if φ ∈ Sn, then φ is even if and only if σφσ−1 is
even.

In cycle notation,let σ=(135)(26)and τ=(456)inS6. (a) Find σ−1.
(b) What is σ(3) equal to? (c) What is the order of σ? (d) Write
down (in cycle notation) the elements of the cyclic subgroup of S6
generated by σ. (e) Compute στσ−1.

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

Let p and q be primes. Prove that pq + 1 is a square if and only
if p and q are twin primes. (Recall p and q are twin primes if p
and q are primes and q = p + 2.) (abstract algebra)

Let
F be a field. Prove that if σ is an isomorphism of F(α1, . . . ,
αn) with itself such that σ(αi) = αi for i = 1, . . . , n, and σ(c)
= c for all c ∈ F, then σ is the identity. Conclude that if E is a
field extension of F and if σ, τ : F(α1, . . . , αn) → E fix F
pointwise and σ(αi) = τ (αi)...

Let G be a finite group and let H, K be normal subgroups of G.
If [G : H] = p and [G : K] = q where p and q are distinct primes,
prove that pq divides [G : H ∩ K].

Let (sn) ⊂ (0, +∞) be a sequence of real numbers. Prove that
liminf 1/Sn = 1 / limsup Sn

Let p,q be prime numbers, not necessarily distinct. If a group G
has order pq, prove that any proper subgroup (meaning a subgroup
not equal to G itself) must be cyclic. Hint: what are the possible
sizes of the subgroups?

Problem 2: (i) Let a be an integer. Prove that 2|a if and only
if 2|a3.
(ii) Prove that 3√2 (cube root) is irrational.
Problem 3: Let p and q be prime numbers.
(i) Prove by contradiction that if p+q is prime, then p = 2 or q
= 2
(ii) Prove using the method of subsection 2.2.3 in our book that
if
p+q is prime, then p = 2 or q = 2
Proposition 2.2.3. For all n ∈...

Let b be a primitive root for the odd prime p. Prove that b^k is
a primitive root for p if and only if gcd(k, p − 1) = 1.

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