2. Groups

Question 1

Group

Answer 1

a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that conditions called group axioms are satisfied

Question 2

4 group axioms

Answer 2

1. Closure: for all x,y in G we have xy in G
2. Associativity: (xy)z=x(yz)
3. Identity Element: there exists an e such that ex=xe=x
4. Inverses: xy=yx=e

Question 3

Abelian Group

Answer 3

A group whose binary operation is commutative.

Question 4

Order of a Group

Answer 4

The number |G| elements in the set G

Question 5

Uniqueness

Answer 5

The indentity element and inverse of x belonging to G is unique

Question 6

Symmetric Group

Answer 6

The set of all permutations of the set X under composition

Question 7

Cyclic Group

Answer 7

there exists an element x in G such that for every y in G there is an m such that y=x^m

Question 8

Example of Cyclic group

Answer 8

The set of invertible elements where m would be -1

Question 9

Generator of Cyclic Group

Answer 9

The letter x such that y=x^m

Question 10

Generator Set

Answer 10

A smaller set of objects, together with a set of operations that can be applied to it that result in a larger collection of objects

Question 11

Dihedral Group

Answer 11

The finite group of symmetries of a regular polygon. It is non abelian and example of connection between groups and geometry.