Group Theory Flashcards
What is a group?
A group is a set, whose elements under a given operation * satisfy the following conditions:
1. Has an identity
2. Is closed under the operation
3. Is associative
4. Has inverse elements for each of its elements
What is the closure property?
A set G is said to have closure (or be closed) under the operation * if for all elements a, b ∈ G:
a * b ∈ G
∈ means “is an element of”
What is Associative property?
For all a, b, c ∈ G, where G is a group under the * operation, the associative property is defined as
a * (b * c) = (a * b) * c
What is identity property of a group?
A set G is said to have an identity property under the binary operation * if for all a, e ∈ G:
a * e = e * a = a
Where e is said to be the identity element of the group.
What is the inverse property of a group?
Say a, e ∈ G, where G is a group under the * operation, if there exists an a-1 such that:
a * a-1 = e
Where e is the identity element, then a-1 is said to be the inverse element.
What is an Abelian group?
An Abelian group is a group whose operations are commutative.
What is an operation?
An operation combines elements of a set in a certain form.
What is a finite group?
A finite group G is a group which has a finite number of elements.
What is a groupoid?
A groupoid is a set P that is closed under the operation *:
a * b ∈ P , for all a, b ∈ P
What is a semi group?
A semi-group is a set P under the operation * which satisfies only the closure and associative properties:
a * (b * c) = (a * b) * c (associative)
for all a, b, c ∈ P.
What is a monoid?
A monoid is a set P under the operation * that has closure, associative properties, and has an identity element e:
a * e = a for all a, e ∈ P.
What is a symmetric group?
Suppose Ω is a finite set. If a bijective mapping, f: Ω→Ω exists, then f is said to be a permutation on Ω; a symmetric group is a group of all permutations on the set Ω.
What is the order of a symmetric group Gn?
|Gn| = n!
