Page 70 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 70
3.2 Basic notions and examples
notes.
I have closely followed Robinson’s book A course in the theory of groups [8] and Ca-
meron’s lecture notes on finite groups [4], thus I claim very little originality as far as for
the exposition goes.
3.2 Basic notions and examples
In this section we collect some basic properties of groups and important examples the
reader should be familiar with in order to read these notes. Most of the proofs in this sec-
tion will be omitted. We will also show how to use GAP in performing explicit calculations
with groups. Concrete examples of computations will be presented.
A convention about the notations. All (or most) of the functions we consider will be
acting from the right. This means that if f : X → Y is a function and x ∈ X , then the image
of x under f will (usually) be denoted by x f or x f .
The main sources of the material covered here are [6] and [8].
3.2.1 Groups
A non-empty set G equipped with a binary operation ◦ is a group if the following hold:
• Associativity: (a ◦ b ) ◦ c = a ◦ (b ◦ c ) for all a ,b, c ∈ G ;
• Identity element: there exists e ∈ G such that e ◦ a = a ◦ e = a for all a ∈ G ;
• Inverse: For every a ∈ G there exists a ∈ G such that a ◦ a = a ◦ a = e .
It is easy to show that the identity element e is uniquely determined, and that every
a ∈ G has a unique inverse, denoted by a −1. For most of the time we write · instead of ◦;
in this case, when there is no confusion, we write 1 instead of e (multiplicative notation).
If g , h ∈ G , we will often use the notation g h = h−1 g h for conjugation of g by h. If the set
G is finite, then we say that G is a finite group, and |G | is called the order of G .
A group G is abelian if a ◦b = b ◦a for all a ,b ∈ G . In this case we often write + instead
of ◦, and the identity element is denoted by 0 (additive notation).
A subset H of G is called a subgroup of G if it is a group under the same operation.
We write H ≤ G . One can verify directly that H is a subgroup of G if and only if a b −1 ∈ H
for all a ,b ∈ H .
If H is a subgroup of G and a ∈ G , then we define left (right) cosets of H by
a H = {a h | h ∈ H },
Ha = {ha | h ∈ H }.
The set of all left cosets of H in G is denoted by G /H , and the set of all right cosets by H \G .
Different left (right) cosets form a partition of G . The number of left (= the number of
notes.
I have closely followed Robinson’s book A course in the theory of groups [8] and Ca-
meron’s lecture notes on finite groups [4], thus I claim very little originality as far as for
the exposition goes.
3.2 Basic notions and examples
In this section we collect some basic properties of groups and important examples the
reader should be familiar with in order to read these notes. Most of the proofs in this sec-
tion will be omitted. We will also show how to use GAP in performing explicit calculations
with groups. Concrete examples of computations will be presented.
A convention about the notations. All (or most) of the functions we consider will be
acting from the right. This means that if f : X → Y is a function and x ∈ X , then the image
of x under f will (usually) be denoted by x f or x f .
The main sources of the material covered here are [6] and [8].
3.2.1 Groups
A non-empty set G equipped with a binary operation ◦ is a group if the following hold:
• Associativity: (a ◦ b ) ◦ c = a ◦ (b ◦ c ) for all a ,b, c ∈ G ;
• Identity element: there exists e ∈ G such that e ◦ a = a ◦ e = a for all a ∈ G ;
• Inverse: For every a ∈ G there exists a ∈ G such that a ◦ a = a ◦ a = e .
It is easy to show that the identity element e is uniquely determined, and that every
a ∈ G has a unique inverse, denoted by a −1. For most of the time we write · instead of ◦;
in this case, when there is no confusion, we write 1 instead of e (multiplicative notation).
If g , h ∈ G , we will often use the notation g h = h−1 g h for conjugation of g by h. If the set
G is finite, then we say that G is a finite group, and |G | is called the order of G .
A group G is abelian if a ◦b = b ◦a for all a ,b ∈ G . In this case we often write + instead
of ◦, and the identity element is denoted by 0 (additive notation).
A subset H of G is called a subgroup of G if it is a group under the same operation.
We write H ≤ G . One can verify directly that H is a subgroup of G if and only if a b −1 ∈ H
for all a ,b ∈ H .
If H is a subgroup of G and a ∈ G , then we define left (right) cosets of H by
a H = {a h | h ∈ H },
Ha = {ha | h ∈ H }.
The set of all left cosets of H in G is denoted by G /H , and the set of all right cosets by H \G .
Different left (right) cosets form a partition of G . The number of left (= the number of
