LEGACY Groups

Axioms of Groups

A Group is a set together with a binary operation denoted by where is a set and is a binary operation.

A Group has to satisfy the following 4 axioms

• Closure: ,
• Identity: s.t
• Inverse: , s.t
• Associativity: ,

A-Level Further Pure 2

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 - *this due that for a fixed , maps onto a unique so every element in a row is unique and since each row/column has different operations with a fixed and each is unique, there is unique elements in the row/column and since there is unique elements in the set each one appears exactly once *
• 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

Modular Arithmetic Groups

You can use modular arithmetic to define a finite group of integers. The operation is defined as The operation is defined as

Groups of Permutations

A group of permutations is a group with a set of permutations of objects with operation composition - the operation of applying permutation then forms a group.

The symmetric group is a group of permutation on elements where the set consists of all possible permutations that can be performed on objects coupled with the operation of composition . (It’s basically a fully complete group of permutations of objects - a normal group of permutations doesn’t have to contain every possible permutation of , These “incomplete” groups will be be subgroups of )

2 Row Notation

You can use 2 row notation to describe permutations like: The top row shows the starting position and denotes what object will end up in the th position and NOT what position the th object will end up in.

Composition For, will be where is found by finding in (in the bottom row) then find in top row of and is the corresponding .

Inverse To find the inverse swap top and bottom.

Groups of Symmetries

You can construct a group of symmetries denoted by by considering a set of all symmetrical transformations of an -sided polygon coupled with the composition operation. There is possible rotations and possible reflections so for an sided polygon there are elements so the group is denoted as meaning ( would be a square)

Cyclic Groups

A cyclic group is a group that can be written where all elements can be written as where is the group generator and . denotes performing the group operation times.

Order

The order of group denoted as is the size of the underlying set The order of an element is the smallest non-negative integer s.t .

Properties

Subgroups

If some subset of the underlying set of a group adheres to the group axioms then it is a subgroup. Every group has at least 2 trivial subgroups and - itself. means is a subgroup of , means is a proper subgroup of meaning ,

Properties

Finite Subgroups Let be a group and a finite subset of the underlying set of ( is a subgroup is closed under the operation of )

Subgroup Generation generates

Lagrange’s Theorem

Isomorphisms

2 groups , are isomorphic if there exists a mapping such that:

• maps all elements of to
• is one to one
• preserves structure:

Properties

If with identity elements , and is an isomorphism then and :

• If has elements of order then has elements of order
• If has subgroups of order , has subgroups of order
• such that

Finding Isomorphism

You can easily find isomorphisms by classifying all possible groups of a given order and their properties

Order	Name	Examples	Properties
1		Trivial group	Only group of order 1
2		under	Only group of order 2
3		under	Only group of order 3
4		under	Cyclic group of order 4
4	Klein four-group ()	Symmetry group of a rectangle	Only non-cyclic group of order 4; Every element (except the identity) has order 2
5		under	Cyclic group of order 5
6		under	Cyclic group of order 6
6	,	Set of all possible permutations of 3 elements, symmetry group of an equilateral triangle	No element of order 6
7		under	Cyclic group of order 7
8		under	Cyclic group of order 8
8		Symmetry group of a square	No element of order 8; Exactly 2 elements of order 4
8			No element of order 8; Exactly 4 elements of order 4
8			No element of order 8; Every element (except the identity) has order 2
8	Quaternion group		No element of order 8; Exactly 6 elements of order 4

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Other Content

Direct Product of Groups

The direct product of two groups and , denoted , is a group defined as follows:

Underlying Set The elements of are ordered pairs such that and . Formally:

Group Operation The operation on is defined component-wise. For , the operation is:

Identity Element The identity element of is:

Inverses: The inverse of an element is:

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