A group consists of a set G and an operation that must satisfy 4 conditions:
Abelian Group - a group that is also commutative.
Subgroup - a subset that is also a group.
Cayley Table - group multiplication table.
Coset - a subset of a group (that is not a group) formed by formed by multiplying a single element of the group by every element of a subgroup.
Left Coset - a Coset formed by multiplying on the left.
Right Coset - a Coset formed by multiplying on the right.
Order - number of elements in a group.
Lagrange's Theorem - the order of a subgroup divides the order of the group.
Homomorphism (similar form) - a function between groups that preserves the group operation:
$${f(x*y) = f(x)*f(y)}$$
Where * is the group operation.
Isomorphism (same form) - a Homomorphism that is also bijective.
Cyclic Group - a group that can be generated by a single element. (always Abelian)
Injective (One-to-One) - no more than 1 element in the domain maps to an element in the codomain so:
if f(a) = f(b) then a = b.
Surjective (Onto) - every element in the codomain has an element mapped to it.
Bijective - function that is both injective and surjective.
Inverses - a function needs to be Bijective to be invertable.
Domain - all possible inputs
Codomain - all possible outputs
Image - the actual outputs of a function. If the image is equal to the Codomain then the function is Surjective.
Preimage - the set of all elements in the domain that map to a subset of the codomain.
To have an equivalence relation 3 conditions must be satisfied:
For a function every element in the domain there must be exactly one corresponding element in the codomain.
Functions must pass the vertical line test on a graph.
If a mapping is not a function then it is called a "Relation".
A relation is a collection of ordered pairs from two sets.
In a relation rather than saying f(a) = b we would say a ~ b.
A Dihedral group is the rotations and reflections of a polygon
The Dihedral group of an n-gon is referred to as \(D_n \)
The order of the Dihedral group is 2n because it has 2n symmetries.
The identity is defined in terms of rotations and reflections: \(s^2 = r^n = e \)
Generally not Abelian except for n = 1 or 2.
But we have a way to apply commutativity: \(sr^k = r^{-k}s \)
$${\text{ord}(sr^k) = 2}$$
This is because \( (sr^k)^2 = e \)
$${ (sr^k)^2 = sr^ksr^k = r^{-k}ssr^k = r^{-k}r^k = e }$$
| Group Name | Notation | Description | Order | Abelian? | Cyclic? |
|---|---|---|---|---|---|
| Integers under addition | ℤ | Infinite cyclic group of integers under addition | ∞ | Yes | Yes |
| Nonzero integers under multiplication | ℤ* | Only ±1 are units (order 2) | 2 | Yes | Yes |
| Integers mod n under addition | ℤn | Cyclic group mod n under addition | n | Yes | Yes |
| Nonzero integers mod p under multiplication | ℤp* | Nonzero elements mod prime p under multiplication | p-1 | Yes | Yes |
| Units mod n under multiplication | Un | Invertible elements mod n under multiplication | φ(n) | Yes | Depends |
| Gaussian integers units under multiplication | ℤ[i]* | ±1, ±i — invertible elements in ℤ[i] | 4 | Yes | Yes |
| Dihedral Group | Dn | The rotations and reflections of a regular polygon | 2n | No for n > 2 | No for n > 2 |