Groups
Table of Contents
1. Essential Concepts
- Binary composition
- A binary composition \(\ast\) on a non-empty set \(G\) is a function from \(G\times G\) into \(G\), i.e., \(\ast:G\times G\mapsto G\) such that for all \(a,b\in G\), we have \(a\ast b \in G\).
- Algebraic structure
- A non-empty set \(G\) together with one or more binary operations is called an algebraic structure. Denoted as \((G,\ast)\)
1.1. Group Axioms
- Associativity
- \[ a\ast (b\ast c) = (a \ast b) \ast c, \forall a,b,c \in G \]
- Existence of Identity
- For every element \(a\in G\), there exists an identity element \(e\in G\) such that \[ a \ast e = e \ast a = a \]
- Existence of Inverse
- For every \(a\in G\), there exists inverse \(a^{-1}\in G\) such that \[ a\ast a^{-1} = a^{-1} \ast a = e \]
If a group is said to be an abelian group or a commutative group if
\[ a\ast b=b\ast a \quad \forall a,b\in G \]
- Finite group
- A group \(G\) is said to be finite if the set \(G\) is finite.
| Structure | Closure | Associativity | Identity | Inverse | Commutative |
|---|---|---|---|---|---|
| Groupoid,Quasi-group | True | ||||
| Semi-group | True | True | |||
| Monoid | True | True | True | ||
| Group | True | True | True | True | |
| Abelian Group | True | True | True | True | True |
1.2. Elementary Properties of Groups
- Identity element is unique
- Inverse of each \( a\in G \) is unique
- \( (a^{-1})^{-1}=a, \forall a \in G \)
- \( (ab)^{-1}=b^{-1}a^{-1}, \forall a,b\in G \)
- \( (a_{1}a_{2}\dots a_{n})^{-1}=a_{n}^{-1} a_{n-1}^{-1}\dots a_{1}^{-1} \)
- Cancellation Laws. \( ab=ac \implies b=c \) and \( ba=ca \implies b=c \)
Proof of Left Cancellation Law. Let \( ab=ac \), then \[ b = eb = (a^{-1}a)b = a^{-1}(ab) = a^{-1}(ac) = (a^{-1}a)c = ec = c. \quad\blacksquare \]
In semi-groups, cancellation laws may not hold.
Prove: A finite semi-group \(S\), in which both the cancellation laws hold, is a group.
Let \(S\) be a finite semi-group in which both the cancellation laws hold.
2. Dihedral Groups
The dihedral groups are the group of symmetries of a regular \(n\)-sided polygon.
Let’s take squares as example, aka \( D_{4} \) or Dihedral group of order 8. The elements include
- Rotation of \( 0\degree \), denoted as \( R_{0} \)
- Rotation of \( 90\degree \), denoted as \( R_{90} \)
- Rotation of \( 180\degree \), denoted as \( R_{180} \)
- Rotation of \( 270\degree \), denoted as \( R_{270} \)
- Reflection about the horizontal axis, denoted as \( F_{H} \)
- Reflection about the vertical axis, denoted as \( F_{V} \)
- Reflection about the main diagonal, denoted as \( F_{D} \)
- Reflection about the other diagonal, denoted as \( F_{D'} \)
We may verify that \( \left\{R_{0}, R_{90}, R_{180}, R_{270}, F_{H}, F_{V}, F_{D}, F_{D'}\right\} \) forms a group, but is not abelian, since \( F_{D'}F_{H} \ne F_{H}F_{D'} \).
Commonly Used Groups
- \(U(n)\)
- A set of positive integers that are \(\in [1,n-1]\) and are co-prime with \(n\) over multiplication.
- \(GL(n, \mathbb{F})\)
- A set of \(n \times n\) invertible matrices over ordinary matrix multiplication, each entry of which is \( \in \mathbb{F} \).
- \(\mathbb{Z}_{n}\)
- A set of integers of \([0,n-1]\) over modulo addition.