1.1 Division (with remainder)
1.1.1: Division with remainder Let a, b ∈ Z with b ̸= 0. Then there exist q, r ∈ Z such that a = qb + r
and 0 ≤ r ≤ |b|. Moreover, these q, r are unique.
1.1.3 Let a, b ∈ Z. We say a divides the integer b, if q ∈ Z exists such that b = qa. This is denoted a|b. In
case no such q exists, we write a ∤ b and we say that a does not divide b
1.1.4 for a, b, c ∈ Z one has:
• If a|b and b|c, then also a|c • 0|a if and only if a = 0
• If a|b and a|c, then also a|(b ± c) • If b ̸= 0 and a|b, then |a| ≤ |b|
• a|0 and 1|a
1.1.5 Let a, b ∈ Z. The greatest common divisor of a and b is defined as the largest integer that is a divisor
of both a and b and is denoted as gcd(a, b). We define gcd(0, 0) = 0. We define the least common multiple
of a and b, denoted lcm(a, b) to be 0 if ab = 0 and to be the smallest positive integer k satisfying both a|k
and b|k otherwise. Two integers a, b are called coprime if gdc(a, b) = 1
1.1.7: The euclidian algorithm The following algorithm computes the gcd in finitely many steps: to
compute gcd(a, b), we rewrite a = bq + r then use that gcd(bq + r, b) =gcd(b, r) until the final gcd can easily
be computed
1.1.9 For a, b, q, r ∈ Z with a = qb + r we have gcd(a, b) =gcd(b, r)
1.1.11 If a < b < 0, then the numbers of divisions with remainder performed by the euclidian algorithm
when determining gcd(a, b) is at most 5 times the number of decimal digits of b.
1.1.12: Bachet-Bézout For a, b ∈ Z there exists integers x, y ∈ Z such that ax + by = gcda, b
1.1.14 Let a, b ∈ Z and put d =gcd(a, b). Every integer that is a divisor of a as well as b is also a divisor of
d. Vice versa, any divisor of d is a common divisor of a and b.
1.1.15 Let a, b ∈ Z. Then, a and b are coprime if and only if there exists x, y ∈ Z such that ax + by = 1
1.1.16 For a, b, c ∈ Z with gcd(a, b) = 1 the following holds: if a|bc, then a|c.
1.2 Prime factorization
1.2.1 A prime number is an integer p > 1 whose only positive divisors are 1 and p.
1.2.3 If p is prime, and a, b ∈ Z such that p|ab, then p|a or p|b.
1.2.4 If p is prime and a1 , . . . , an are integers such that p|a1 . . . an , then there exists an index 1 ≤ k ≤ n
such that p|ak .
1.2.5: unique prime factorization Every integer greater than 1 can be written as a product of the primes.
This product is unique up to the order of the factors.
1.2.7: Euclid There exists infinitely many primes.
1.2.9 If p is prime and a ∈ Z is not equal to 0 or ±1, then we write vp (a) for the number of times p appears
in the prime factorization of |a|. Moreover, we put vp (1) = vp (−1) and vp (0) = ∞.
1.2.10 Let a, b be integers.
1. For every prime p we have vp (ab) = vp (a) + vp (b)
1
, 2. We have a|b if and only if every prime p satisfies vp (a) ≤ vp (b)
3. If a and b are not both 0, then
Y
gcd(a, b) = pmin{vp (a),vp (b)}
p prime
if a ̸= 0 and also b ̸= 0, then
Y
lcm(a, b) = pmax{vp (a),vp (b)}
p prime
If a|c and b|c, then lcm(a, b)|c
gcd(a, b)·lcm(a, b) = |ab|
3.1 Groups
3.1.1 A group is a triple (G, ·, e) where G is a set, e ∈ G and · is a map from G × G to G which we write as
(x, y) 7→ x · y, satisfying:
1. (assosicativity) For all x, y, z ∈ G we have (x · y) · z = x · (y · z)
2. (unit element) For all x ∈ G we have e · x = x = x · e
3. (inverses) For all x ∈ G a y ∈ G exists such that x · y = e = y · x
Moreover, a group is called commutuative or abelian if it is also holds that for all x, y ∈ G we have x·y = y ·x.
3.1.6 Let (G, ·, e) be a group.
1. If e′ ∈ G satisfies e′ x = x or xe′ = x for some x ∈ G, then e′ = e
2. For every x ∈ G there is precisely one y ∈ G with xy = e = yx
3. For any fixed a ∈ G, the mao λa : G → G; x 7→ ax is a bijection from G to itself. Similarly,
ρa : G → G; x 7→ xa is a bijection.
3.1.7 Let (G, ·, e) be a group and x ∈ G The element y ∈ G such that xy = e = yx is called the inverse of x
in G. It is denoted by x−1, by Theorem 3.1.6 x−1 is unique and hence well-defined. In case of an abelian
group G with group law denoted as +, this inverse element is called the opposite of x in G, and it is denoted
as x.
3.1.9 Let G be a group and let a, a1 , a2 , . . . , an ∈ G. Then we have
1. (a−1 )−1 = a
−1 −1
2. (a1 . . . an )−1 = a−1
n · an−1 . . . a1
3. (an )−1 = (a−1 )
3.1.10: The multiplication table One can describe a group G consisting of only finitely many elements
by means of a table which contains all results of multiplying pairs of elements of G. We represent this in a
2
,matrix (ai,j ). Position a1,1 remains empty, or we could write the name of the group here. In the remainder
of the first row we write the elements of G, and the same in the first column. In position ai,j (with i, j ≥ 2)
we put the product ai,1 · a1,j . (The product with an element from column 1 on the left, and an element from
row 1 on the right! the order obviously makes a difference in the case of non-abelian groups.)
Lecture 1: 5-9
Group theory 2022
Quizzes will not be difficult at all, 2 or 3 questions
Application: algebra, complex analysis, and more
An example
1. Rotate by 120 degrees clockwise (called ρ)
2. Rotate by 240 degrees clockwise (called ρ2 )
3. Rotate by 360 degrees clockwise (identity operation)
4. Reflect along the axis coming from the top corner/vertical line dividing the base side into two pieces
(called τ1 )
5. Reflect along the axis coming from the left vertex (called τ2 )
6. Reflect along the axis coming from the right vertex (called τ3 )
Composition
If you first apply ρ and then τ1 , you end up with τ2
If you first apply τ2 and then apply τ3 , we end up with ρ2 .
We call a set closed with respect to an operation if applying the operation to elements of the set always
results in another element in the set Exercise: Show that G = {id, ρ, ρ2 , τ1 , τ2 , τ3 } is closed with respect to
composition operation
Unit element A composition with the unit element, nothing happens (id ◦ ϕ = ϕ and ϕ ◦ id = ϕ)
Inverses Can you compose a map with another map so that the composition map is the identity map? Yes,
the list of inverses is in the notes.
ρ2 ◦ ρ means you first apply ρ and then ρ2 .
3.1 groups Definition of a group. Exercise: (f ◦ g) ◦ h = f ◦ (g ◦ h) holds for the set G and prove that G is
a group. Is it abelian? (no, see notes)
A group is closed by definition
Set of integers Is (Z, +, 0) a group? Yes. Check:
G0: a, b ∈ Z, then (a + b) ∈ Z G1: (a + b) + c = a + (b + c) for all a, b, c ∈ Z
G2: a + 0 = 0 + a = a for all a ∈ Z
G3: a + (−a) = 0 = (−a) + a for all a ∈ Z
G4: a + b = b + a for all a, b ∈ Z
Thus this is a commutative group.
Next example: (Z, ·, 1) is not a group because there are no inverses. If we change Z to Q it is still not a
group because 0 does not have an inverse. If we change Q to Q\{0}, it is a group.
(Z, −, 0) is also not a group because - is not associative Vector spaces (V, +, 0) is an abelian group.
Group or not? 1. is not a group (counter example 1 + 2 = 3 ∈
/ S)
3
, 2. is not a group (2 · 2 = 4 ∈
/ S, and no inverse elements.) 3. Is a group, because it satisfies the axioms.
However, it is not abelian.
Some definitions and remarks See handouts.
elementary properties 1. The identity element is unique
Take y to be th inverse element of x. Then, e′ x = x =⇒ e′ xy = xy =⇒ e′ = e Thus it is unique.
2. The inverse element is unique
Let y be an inverse of x. Suppose that y ′ satisfies xy ′ = e. Then, y = ye = yxy ′ = y ′ since xy = yx = e.
Thus we conclude that y = y ′ .
3. For any fixed a ∈ G, the map λa : G → G; x 7→ ax is a bijection from G to itself. Similarly,
ρa : G → G; x → xa is a bijection.
λa : G → G; x 7→ ax.
Injective: λa (x) = λa (x′ ) =⇒ x = x′ .λa (x) = ax = ax′ = λ( x′ ). Let b be the inverse of a. Then,
bax = bax′ =⇒ x = x′
Surjective: ∀y ∈ G ∃x ∈ G : λa (x) = y. Let b be the inverse of a. Take x = by =⇒ λa (by) = aby = y
Since λa is both injective and surjective, it is bijective.
Exercise: show for ρa .
Some more proofs
1. (a−1 )−1 = a
(a−1 )−1 · a−1 · a = e · a =⇒ (a−1 )−1 = a
−1 −1
2. (a1 . . . an )−1 = a−1
n an−1 . . . a1
−1 −1
(a−1 −1 1 n
n an−1 . . . a1 )(a1 . . . an ) = an an . . . a1 a1 = e = e
3. (an )−1 = (a−1 )n Take ai = a.
Multiplication table The group is commutative iff the table is symmetric along the diagonal axis passing
through the upper left to the lower right corner.
Start with the side, end with the top.
2.1 Residue classes modulo N
2.1.1 Two integers a, b are called congruent modulo N if N |a − b. This is denoted by a ≡ b mod N . We call
N the modulus.
2.1.2 Let a, b, c be integers. Then the following assertions hold.
1. (Reflexivity) We have a ≡ a mod N
2. (Symmetry) We have a ≡ b mod N if and only if b ≡ a mod N
3. (Transitivity) If a ≡ b mod N and b ≡ c mod N , then a ≡ c mod N
2.1.3 For a ∈ Z the residue class of a modulo N is defined as {b ∈ Z | b ≡ a mod N }.
¯
If the N is clear, we can also write ā. Example: if N = 4 then 1̄ = 17.
2.1.5 For a, b ∈ Z one has a mod N = b mod N if and only if a ≡ b mod N .
2.1.6 Let a¯1 , a¯2 , b¯1 , b¯2 be residue classes modulo N , where a1 , a2 , b1 , b2 are integers. Suppose that a¯1 = a¯2
4
1.1.1: Division with remainder Let a, b ∈ Z with b ̸= 0. Then there exist q, r ∈ Z such that a = qb + r
and 0 ≤ r ≤ |b|. Moreover, these q, r are unique.
1.1.3 Let a, b ∈ Z. We say a divides the integer b, if q ∈ Z exists such that b = qa. This is denoted a|b. In
case no such q exists, we write a ∤ b and we say that a does not divide b
1.1.4 for a, b, c ∈ Z one has:
• If a|b and b|c, then also a|c • 0|a if and only if a = 0
• If a|b and a|c, then also a|(b ± c) • If b ̸= 0 and a|b, then |a| ≤ |b|
• a|0 and 1|a
1.1.5 Let a, b ∈ Z. The greatest common divisor of a and b is defined as the largest integer that is a divisor
of both a and b and is denoted as gcd(a, b). We define gcd(0, 0) = 0. We define the least common multiple
of a and b, denoted lcm(a, b) to be 0 if ab = 0 and to be the smallest positive integer k satisfying both a|k
and b|k otherwise. Two integers a, b are called coprime if gdc(a, b) = 1
1.1.7: The euclidian algorithm The following algorithm computes the gcd in finitely many steps: to
compute gcd(a, b), we rewrite a = bq + r then use that gcd(bq + r, b) =gcd(b, r) until the final gcd can easily
be computed
1.1.9 For a, b, q, r ∈ Z with a = qb + r we have gcd(a, b) =gcd(b, r)
1.1.11 If a < b < 0, then the numbers of divisions with remainder performed by the euclidian algorithm
when determining gcd(a, b) is at most 5 times the number of decimal digits of b.
1.1.12: Bachet-Bézout For a, b ∈ Z there exists integers x, y ∈ Z such that ax + by = gcda, b
1.1.14 Let a, b ∈ Z and put d =gcd(a, b). Every integer that is a divisor of a as well as b is also a divisor of
d. Vice versa, any divisor of d is a common divisor of a and b.
1.1.15 Let a, b ∈ Z. Then, a and b are coprime if and only if there exists x, y ∈ Z such that ax + by = 1
1.1.16 For a, b, c ∈ Z with gcd(a, b) = 1 the following holds: if a|bc, then a|c.
1.2 Prime factorization
1.2.1 A prime number is an integer p > 1 whose only positive divisors are 1 and p.
1.2.3 If p is prime, and a, b ∈ Z such that p|ab, then p|a or p|b.
1.2.4 If p is prime and a1 , . . . , an are integers such that p|a1 . . . an , then there exists an index 1 ≤ k ≤ n
such that p|ak .
1.2.5: unique prime factorization Every integer greater than 1 can be written as a product of the primes.
This product is unique up to the order of the factors.
1.2.7: Euclid There exists infinitely many primes.
1.2.9 If p is prime and a ∈ Z is not equal to 0 or ±1, then we write vp (a) for the number of times p appears
in the prime factorization of |a|. Moreover, we put vp (1) = vp (−1) and vp (0) = ∞.
1.2.10 Let a, b be integers.
1. For every prime p we have vp (ab) = vp (a) + vp (b)
1
, 2. We have a|b if and only if every prime p satisfies vp (a) ≤ vp (b)
3. If a and b are not both 0, then
Y
gcd(a, b) = pmin{vp (a),vp (b)}
p prime
if a ̸= 0 and also b ̸= 0, then
Y
lcm(a, b) = pmax{vp (a),vp (b)}
p prime
If a|c and b|c, then lcm(a, b)|c
gcd(a, b)·lcm(a, b) = |ab|
3.1 Groups
3.1.1 A group is a triple (G, ·, e) where G is a set, e ∈ G and · is a map from G × G to G which we write as
(x, y) 7→ x · y, satisfying:
1. (assosicativity) For all x, y, z ∈ G we have (x · y) · z = x · (y · z)
2. (unit element) For all x ∈ G we have e · x = x = x · e
3. (inverses) For all x ∈ G a y ∈ G exists such that x · y = e = y · x
Moreover, a group is called commutuative or abelian if it is also holds that for all x, y ∈ G we have x·y = y ·x.
3.1.6 Let (G, ·, e) be a group.
1. If e′ ∈ G satisfies e′ x = x or xe′ = x for some x ∈ G, then e′ = e
2. For every x ∈ G there is precisely one y ∈ G with xy = e = yx
3. For any fixed a ∈ G, the mao λa : G → G; x 7→ ax is a bijection from G to itself. Similarly,
ρa : G → G; x 7→ xa is a bijection.
3.1.7 Let (G, ·, e) be a group and x ∈ G The element y ∈ G such that xy = e = yx is called the inverse of x
in G. It is denoted by x−1, by Theorem 3.1.6 x−1 is unique and hence well-defined. In case of an abelian
group G with group law denoted as +, this inverse element is called the opposite of x in G, and it is denoted
as x.
3.1.9 Let G be a group and let a, a1 , a2 , . . . , an ∈ G. Then we have
1. (a−1 )−1 = a
−1 −1
2. (a1 . . . an )−1 = a−1
n · an−1 . . . a1
3. (an )−1 = (a−1 )
3.1.10: The multiplication table One can describe a group G consisting of only finitely many elements
by means of a table which contains all results of multiplying pairs of elements of G. We represent this in a
2
,matrix (ai,j ). Position a1,1 remains empty, or we could write the name of the group here. In the remainder
of the first row we write the elements of G, and the same in the first column. In position ai,j (with i, j ≥ 2)
we put the product ai,1 · a1,j . (The product with an element from column 1 on the left, and an element from
row 1 on the right! the order obviously makes a difference in the case of non-abelian groups.)
Lecture 1: 5-9
Group theory 2022
Quizzes will not be difficult at all, 2 or 3 questions
Application: algebra, complex analysis, and more
An example
1. Rotate by 120 degrees clockwise (called ρ)
2. Rotate by 240 degrees clockwise (called ρ2 )
3. Rotate by 360 degrees clockwise (identity operation)
4. Reflect along the axis coming from the top corner/vertical line dividing the base side into two pieces
(called τ1 )
5. Reflect along the axis coming from the left vertex (called τ2 )
6. Reflect along the axis coming from the right vertex (called τ3 )
Composition
If you first apply ρ and then τ1 , you end up with τ2
If you first apply τ2 and then apply τ3 , we end up with ρ2 .
We call a set closed with respect to an operation if applying the operation to elements of the set always
results in another element in the set Exercise: Show that G = {id, ρ, ρ2 , τ1 , τ2 , τ3 } is closed with respect to
composition operation
Unit element A composition with the unit element, nothing happens (id ◦ ϕ = ϕ and ϕ ◦ id = ϕ)
Inverses Can you compose a map with another map so that the composition map is the identity map? Yes,
the list of inverses is in the notes.
ρ2 ◦ ρ means you first apply ρ and then ρ2 .
3.1 groups Definition of a group. Exercise: (f ◦ g) ◦ h = f ◦ (g ◦ h) holds for the set G and prove that G is
a group. Is it abelian? (no, see notes)
A group is closed by definition
Set of integers Is (Z, +, 0) a group? Yes. Check:
G0: a, b ∈ Z, then (a + b) ∈ Z G1: (a + b) + c = a + (b + c) for all a, b, c ∈ Z
G2: a + 0 = 0 + a = a for all a ∈ Z
G3: a + (−a) = 0 = (−a) + a for all a ∈ Z
G4: a + b = b + a for all a, b ∈ Z
Thus this is a commutative group.
Next example: (Z, ·, 1) is not a group because there are no inverses. If we change Z to Q it is still not a
group because 0 does not have an inverse. If we change Q to Q\{0}, it is a group.
(Z, −, 0) is also not a group because - is not associative Vector spaces (V, +, 0) is an abelian group.
Group or not? 1. is not a group (counter example 1 + 2 = 3 ∈
/ S)
3
, 2. is not a group (2 · 2 = 4 ∈
/ S, and no inverse elements.) 3. Is a group, because it satisfies the axioms.
However, it is not abelian.
Some definitions and remarks See handouts.
elementary properties 1. The identity element is unique
Take y to be th inverse element of x. Then, e′ x = x =⇒ e′ xy = xy =⇒ e′ = e Thus it is unique.
2. The inverse element is unique
Let y be an inverse of x. Suppose that y ′ satisfies xy ′ = e. Then, y = ye = yxy ′ = y ′ since xy = yx = e.
Thus we conclude that y = y ′ .
3. For any fixed a ∈ G, the map λa : G → G; x 7→ ax is a bijection from G to itself. Similarly,
ρa : G → G; x → xa is a bijection.
λa : G → G; x 7→ ax.
Injective: λa (x) = λa (x′ ) =⇒ x = x′ .λa (x) = ax = ax′ = λ( x′ ). Let b be the inverse of a. Then,
bax = bax′ =⇒ x = x′
Surjective: ∀y ∈ G ∃x ∈ G : λa (x) = y. Let b be the inverse of a. Take x = by =⇒ λa (by) = aby = y
Since λa is both injective and surjective, it is bijective.
Exercise: show for ρa .
Some more proofs
1. (a−1 )−1 = a
(a−1 )−1 · a−1 · a = e · a =⇒ (a−1 )−1 = a
−1 −1
2. (a1 . . . an )−1 = a−1
n an−1 . . . a1
−1 −1
(a−1 −1 1 n
n an−1 . . . a1 )(a1 . . . an ) = an an . . . a1 a1 = e = e
3. (an )−1 = (a−1 )n Take ai = a.
Multiplication table The group is commutative iff the table is symmetric along the diagonal axis passing
through the upper left to the lower right corner.
Start with the side, end with the top.
2.1 Residue classes modulo N
2.1.1 Two integers a, b are called congruent modulo N if N |a − b. This is denoted by a ≡ b mod N . We call
N the modulus.
2.1.2 Let a, b, c be integers. Then the following assertions hold.
1. (Reflexivity) We have a ≡ a mod N
2. (Symmetry) We have a ≡ b mod N if and only if b ≡ a mod N
3. (Transitivity) If a ≡ b mod N and b ≡ c mod N , then a ≡ c mod N
2.1.3 For a ∈ Z the residue class of a modulo N is defined as {b ∈ Z | b ≡ a mod N }.
¯
If the N is clear, we can also write ā. Example: if N = 4 then 1̄ = 17.
2.1.5 For a, b ∈ Z one has a mod N = b mod N if and only if a ≡ b mod N .
2.1.6 Let a¯1 , a¯2 , b¯1 , b¯2 be residue classes modulo N , where a1 , a2 , b1 , b2 are integers. Suppose that a¯1 = a¯2
4
