Groups Flashcards
What properties does a group need to have?
- Closure
- Associativity
- Identity
- Inverses
Define Closure:
∀g,h ∈ G, g * h ∈ G
(* signifies the group operation, not multiplication)
For example, the set of even integers is closed under addition because the sum of any two even integers is always even.
Define Associative:
∀g,h,k ∈ G, (g * h) * k = g * (h * k)
(* signifies the group operation, not multiplication)
For example, addition is associative because for any real numbers g,h,k:
(g + h) + k = g + (h + k)
Define Identities:
∃e ∈ G | ∀g ∈ G, e * g = g* e = g
(* signifies the group operation, not multiplication. e = Identity)
An identity for an operation is an element that, when combined with any other element using that operation, leaves the other element unchanged.
’+’ Identity is 0 as a + 0 = a
‘x’ Identity is 1 as a x 1 = a
Define Inverses:
∀a ∈ G, ∃b ∈ G | a * b = b* a = identity
(* signifies the group operation, not multiplication)
x * x’ = e
For a given operation and its identity element, the inverse of an element a is another element b such that when a and b are combined using the operation, the result is the identity element.
Let (G, *) be a group.
is the empty set a subgroup of G?
No. The empty set is a subset of G, but it is NOT a subgroup
What defines a Subgroup.
Given a group G with an operation ∘, a subset H of G is a subgroup if:
H is non-empty.
H is closed under the operation ∘. (∀g,h ∈ H, g ∘ h ∈ H)
If e is the identity for (G, ∘), the e ∈ H.
For every element a ∈ H, its inverse a^(−1) ∈ H.
A subgroup retains the group structure but might have fewer elements than the original group.
If the trivial subgroup of G is {e}, what does that tell us?
e is the identity of group G
How does Raising the Powers of Elements work (mechanically)?
In the context of groups, given an element a and an integer n:
- a^n denotes the product of a with itself n times.
- If n > 0, a^n = a ∘ a ∘ … ∘ a (n times).
- a^0 is the identity element of the group.
- a^(-n) is the inverse of a^n.
It must be noted the operation ∘ will be whatever the operation on the group is. It is not simply a multiple.
Why Raise the Powers of Elements in Group Theory?
- Cyclic Groups: Powers help determine if a group is cyclic. If an element’s powers can generate all group members, the group is cyclic.
- Order of Elements: The smallest positive power that gives the identity element is the element’s order. It’s a key property in group studies.
- Group Structure: Powers can reveal patterns and symmetries, helping understand the group’s structure and properties.
Raising powers is a fundamental operation that provides insights into the behavior and properties of groups.
Determine < 3 > in the group (ℤ₁₁ - {0}, x)
- 3^0 = 1 (Identity)
- 3^1 = 3
- 3^2 = 9
- 3^3 = 27 = 5
- 3^4 = 81 = 4
- 3^5 = 243 = 1
(At this point, we’ve reached the identity element, and will repeat)
< 3 > in the group (ℤ₁₁ - {0}, x) = {1,3,9,5,4}
Two groups (G,*) and (H,∘) are isomorphic if and only if ..?
Two groups (G,*) and (H,∘) are isomorphic if and only if there exists a bijection f: G ➔ H such that for all x, y ∈ G, f (x * y) = f (x) ∘ f (y).
What can be an easier way to find an isomorphism/bijection?
The Schröder-Bernstein theorem:
-
Find an Injective Function from A to B:
Define a function f: A → B and show it’s injective. -
Find an Injective Function from ( B ) to ( A ):
Define a function g: B → A and show it’s injective. -
Apply Schröder-Bernstein:
If you’ve successfully shown the two injective functions as described, then by the Schröder-Bernstein theorem, a bijection h: A → B exists.
A group is Abelian iff..?
The groups binary operation is commutative.
A binary operation * on S is commutative if a * b = b * a, ∀a,b ∈ S
A binary operation * on S is commutative if..?
A binary operation * on S is commutative if:
a * b = b * a, ∀a,b ∈ S
