Axioms of Groups
A Group is a set together with a binary operation denoted by
A Group has to satisfy the following 4 axioms. Other fundamental corollaries denoted by
Closure: , Associativity: , Inverse: , s.t . Works both ways (Directly implied by axioms 1-4) Inverses are unique.
Identity: s.t . Works both ways (Directly implied by axioms 1-4) Identity is unique.
Note that commutativity is not required
Corollary Proofs
The following properties are a direct result of the axioms and are sometimes included in the definition but do not have to be.
Proof
- Note that
where . All these elements are guaranteed to exist by . - Consider
. By . Also by using the equivalent statements of we get . So we get that - Substituting in we get
Proof
- Consider
. . Using and . Using and - Hence
Proof
- Assume that that exists two distinct inverses of
such that - Then
Proof
- Assume that there exists two distinct identity elements such that
- Then
Proof
- Assume that for a given
Proof
Cayley Tables
A Cayley Table fully describes a finite group by showing all possible products of elements in the group Example:
When a group’s elements are displayed in a Cayley Table:
- All entries are
- Every entry appears exactly once in each row and column
- Trivial with
- Trivial with
must appear in every row and column and it’s position is symmetric about the leading diagonal - Since
(and can only be formed from this operation), when is in location it has to be in as or are or so
- Since
Order
The order of group
Properties
and
Subgroups
If some subset of the underlying set of a group adheres to the group axioms then it is a subgroup. Every group
means is a subgroup of , means is a proper subgroup of meaning ,
Let
Subgroup Generation
Lagrange’s Theorem
Isomorphisms
2 groups
maps all elements of to is one to one preserves structure:
Properties
If
- If
has elements of order then has elements of order - If
has subgroups of order , has subgroups of order such that
Types of Groups
Modular Arithmetic Groups
You can use modular arithmetic to define a finite group of integers.
The operation
Groups of Permutations
A group of Permutations is a group with a set of permutations of
The symmetric group
2 Row Notation
You can use 2 row notation to describe permutations like:
Composition
For
Groups of Symmetries
You can construct a group of symmetries denoted by
Cyclic Groups
A cyclic group is a group that can be written where all elements can be written as
Direct Product of Groups
The direct product of two groups
Underlying Set
The elements of
Group Operation
The operation on
Identity Element
The identity element of
Inverses:
The inverse of an element