Y1 Group Theory

Binary operation on a set

An operation that produces another member of the set

Another word for binary operation

Closed

Identity element

Element in the set

a * e = a and

e * a = a

(any operation will produce the original element)

Inverse element

Element in the set

Satisfies a * b = e and

b * a = e

What is the identity element signified by

e

Associative

(a * b) * c = a * (b * c)

Group

A set and binary operation that satisfy the axioms:

-Closed

-Identity Element

-Inverse

-Associative

Which type of group can Cayley tables be used for

Finite groups

How to construct a Cayley table

The operation in the top left corner, with every element at the start of a column and row

Cayley Table rules

-All entries must be members of the group

-Every element appears exactly once in each column and row

-The identity elements are symmetric across the leading diagonal

Identity permutation

All elements remain in the same position

How to find the two row notation of a permutation

Each element is moved from its position in the first row to its position in the second row

How too find the composition of two permutations (a∘b)

Look at where the element in position 1 of b moves to. Then look at where the item in this new position ends up in a and write that below position 1 of your new composition. Repeat for the other positions

How to find the inverse of a composition

Read from the bottom row to the top row

Group of symmetry

Contains the permutation for every possible reflectional and rotational symmetry

Cyclic group

A group where an element can generate the entire group

Generator

Repeatedly applying the operation to the same element, to create other members of the group

What is the order of a group

The number of elements it has

|G|

The order of group G

The order of an element

The lowest positive integer k such that a^k = e

How can you use order to determine whether a group is cyclic

If any element has an order that equals the order of the group

If an element a has finite order n, then state what is known about a^m

n|m

Subgroup

If some subset of the underlying set forms a group under the same operation and group axioms

H ⊂ G

H is a proper subgroup of G

H ⊆ G

H is a subgroup of G

Which subgroups at minimum do all groups have

-The group itself

-The trivial subgroup

Trivial subgroup

The subgroup containing only the identity element

Proper subgroup

Any subgroup that is not the group itself

How to find subgroups

Use elements in the group to generate them

Langrange’s theorem

If H is a subgroup of G, then |H| divides |G|