Exercise 4.1
Given a,b,c ∈ N \ {0}, show that .
Exercise 4.2
Show that
(i) There exist infinitely many primes of the form 3n + 2, n ∈ N. (ii) There exist infinitely many primes of the form 6n + 5, n ∈ N.
Exercise 4.3
The numbers Fn = 22n + 1 are called the Fermat numbers.
(i) Show that gcd(Fn,Fn+1) = 1, n ∈ N.
(ii) Use (i) to show that there are infinitely many primes.
Exercise 4.4 Show that
(i) If a is even and b is odd, then gcd(a,b) = gcd(a/2,b).
(ii) If both a and b are even, then gcd(a,b) = 2gcd(a/2,b/2).
Exercise 4.5
Find all x,y ∈ Z such that
(i) 56x + 72y = 39,
(ii) 84x − 439y = 156.
Exercise 4.6
Given a group G = (S,·), where S is the underlying set, and · is the groups law. Define a new function
⊠ : S × S → S
(a,b) 7→ a ⊠ b := b · a
Show that (S,⊠) is a group.
Exercise 4.7
Given a group G, show that
(i) If the order of every nonidentity element of G is 2, then G is Abelian.
(ii) If a,b ∈ G, then |ab| = |ba|, i.e., ab and ba have the same order.
Exercise 4.8
Given f : (R,+) → (C \ {0},×), x 7→ eix.
(i) Show that f is a homomorphism.
(ii) Find kerf.
(iii) Find imf.
Exercise 4.9
Given groups G, G′, and f : G → G′ a surjective homomorphism. Show that (i) (2pts) G′ is cyclic if G is cyclic.
(ii) (2pts) G′ is abelian if G is abelian.
Exercise 4.10
Given group G and a function f : G → G, x 7→ x−1. Show that the following are equivalent,
(a) G is abelian.
(b) f is a homomorphism.
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Exercise 4.11
Show that {1,(12)(34),(13)(24),(14)(23)} is a subgroup of A4.
Exercise 4.12
Given group G with |G| even, show that G contains an element of order 2.
Exercise 4.13
(i) Show that the normal subgroup property is not transitive.
(ii) Show that a subgroup of index 2 is normal.
(iii) Show that a subgroup of index 3 is not necessarily normal.
Exercise 4.14
Let G be a group of order p2, with p prime. Show that (i) G has at least one subgroup of order p.
(ii) If G contains only one subgroup of order p, then G is cyclic.
Exercise 4.15
State a converse of Lagrange’s theorem. If the statement is true, find a reference, otherwise provide a counterexample.
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