Exercise 4.1
Given a, b, c ∈ N \ {0}, show that a | bc iff a c. gcd(a,b)
Exercise 4.2
Show that
(i) (2 pts) There exist infinitely many primes of the form 3n + 2, n ∈ N.
(ii) (2 pts) There exist infinitely many primes of the form 6n + 5, n ∈ N. Exercise 4.3 (4 pts)
The numbers Fn = 22n + 1 are called the Fermat numbers. (i) (2 pts) Show that gcd(Fn, Fn+1) = 1, n ∈ N.
(ii) (2 pts) Use (i) to show that there are infinitely many primes. Exercise 4.4 (2 pts)
Show that
(i) (1 pt) If a is even and b is odd, then gcd(a, b) = gcd(a/2, b).
(ii) (1 pt) If both a and b are even, then gcd(a, b) = 2gcd(a/2, b/2). Exercise 4.5 (4 pts)
Find all x, y ∈ Z such that
(i) (2pts) 56x+72y=39,
(ii) (2 pts) 84x − 439y = 156. Exercise 4.6 (2 pts)
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) → a ⊠ b := b · a
Show that (S, ⊠) is a group. Exercise 4.7 (4 pts)
Given a group G, show that
(i) (2 pts) If the order of every nonidentity element of G is 2, then G is Abelian.
(ii) (2 pts) If a, b ∈ G, then |ab| = |ba|, i.e., ab and ba have the same order. Exercise 4.8 (6 pts)
Given f : (R, +) → (C \ {0}, ×), x → eix.
(i) (2 pts) Show that f is a homomorphism.
(ii) (2pts) Find kerf. (iii) (2pts) Find imf.
Exercise 4.9 (4 pts)
Given groups G, G′, and f : G → G′ a surjective homomorphism. Show that (i) (2 pts) G′ is cyclic if G is cyclic.
(ii) (2 pts) G′ is abelian if G is abelian. Exercise 4.10 (2 pts)
Given group G and a function f : G → G, x → x−1. Show that the following are equivalent,
(a) G is abelian.
(b) f is a homomorphism.
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Exercise 4.11 (2 pts)
Show that {1, (12)(34), (13)(24), (14)(23)} is a subgroup of A4 . Exercise 4.12 (2 pts)
Given group G with |G| even, show that G contains an element of order 2.
Exercise 4.13 (6 pts)
(i) (2 pts) Show that the normal subgroup property is not transitive. (ii) (2 pts) Show that a subgroup of index 2 is normal.
(iii) (2 pts) Show that a subgroup of index 3 is not necessarily normal. Exercise 4.14 (4 pts)
Let G be a group of order p2, with p prime. Show that (i) (2 pts) G has at least one subgroup of order p.
(ii) (2 pts) If G contains only one subgroup of order p, then G is cyclic. Exercise 4.15 (2 pts)
State a converse of Lagrange’s theorem. If the statement is true, find a reference, otherwise provide a counterexample.
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