1) Groups and Symmetry Flashcards
What is a binary operation
A binary operation on a set G is a function ∗ : G x G ⇥ G
What does it mean to be associative
A binary operation ∗ on a set G is said to be associative if, for any three elements a, b, and c in
G, (a∗b)∗c=a∗(b∗c)
What is the identity and inverse of a set
Let G be a set with binary operation ∗
An element e ∈ G is said to be an identity element with respect to ∗ if for all a , we have a ∗ e = a and e ∗ a = a.
a ∈ G has inverse a0 ∈ G if a ∗ a0 = e and a0 ∗ a = e.
How many identity elements can there be for a set equipped with a binary operation
At most one
In the context of associative operations, what is guaranteed about the inverses of elements
For associative operations, each element has at most one inverse
What is a group
A group is a pair (G, ∗), where -
* G is a non-empty set;
* ∗: G x G ⇥ G is an associative binary operation
* There exists an identity element e ∈ G
* Each g ∈ G has an inverse g’ ∈ G
What are the key examples of groups
- Integers under addition
- Symmetric Groups under composition
- Isometries under compositon
- Invertiable matrices under matrix multiplication
- Vector Space under addition
What is a subgroup
We say that (H, ∗) is a subgroup of (G, ∗), if -
* For all g, h ∈ H we have g ∗ h ∈ H
* (H, ∗) is a group
What is an abelian group
A group those pairs of elements are commutative,
∀g, h ∈ G we have g ∗ h = h ∗ g
What do we know about the multiplication table of abelian groups
The table will be equal to its transpose
What is an isomorphism
An isomorphism between two groups,
(G,∗) and(H,∘), is defined by a bijective function ϕ:G→H such that ∀ a,b ∈ G, ϕ(a∗b)=ϕ(a)∘ϕ(b)
How do we denote two groups are isomorphic
G ≅ H
What are the properties of groups
- The identity element is its own inverse, that is, e^-1 = e
- For all g ∈ G, we have (g^-1)^-1 = g
- For all g, h ∈ G, we have (gh)^-1 = h^-1g^-1
- (Left cancellation) For all a, b, c ∈ G, if ab = ac then b = c
- (Right cancellation) For all a, b, c ∈ G, if ba = ca then b = c
