Extra Pure Flashcards
What is a group?
A set of elements with a binary operation
What is a binary operation?
An operation with 2 inputs and 1 output
What are the 4 group axioms?
Identity
Closure
Inverse
Associativity
What does the Identity axiom mean?
There is an element that doesn’t change other elements when composed with them
What does the Inverse axiom mean?
For every element in the group there is an element when composed together, form the identity
WHat does the Closure axiom mean?
If you compose 2 elements, you get another one in the set
What does the Associativity axiom mean?
(a . b) . c = a. (b . c)
What is an Abelian group?
If a group is said to be commutative
What is the order of a group?
The number of elements in the group
What does it mean for a group to be isomorphic?
- If they are the same group
- If there is a mapping between the 2 groups
How would you know if 2 groups are isomorphic?
If the order of the elements and the group are the same
What is a cyclic group?
A group generated by 1 element in the group
Represented by where e is the element that generates the group
What group is guaranteed to be cyclic?
If the order of the group is prime so every non-identity element is a generator
What makes group H a subgroup of G?
- The elements of H are a subset of the elements of G
- They have the same binary operator
- It is a group
A fact of subgroups
The order of the subgroups divides the order of the groups
