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 |
$${(ab)^{-1} = b^{-1} a^{-1} }$$
$${(ab)^{2} = a^2 b^2 \Rightarrow ab = ba }$$
$${\text{ord}(g) = 2 \; \forall \; g \in G \Rightarrow \text{G is Abelian} }$$
The Symmetric Group is an example of the Permutation group.
It is the group of all permutations of n elements.
The most common example is: \(S_n = {1, 2, 3 \cdots n} \)
The order of the group \(S_n \) is just all permutations so n!.
The Symmetric Group is bijective but it's not commutative (not Abelian)
Every permutation can be written as the product of disjoint cycles.
The order of an n-cycle is n.
The order of disjoint cycles is the LCM of the length of all cycles.
The order of an n-cycle raised to a power of k:
$${\text{ord}\left( n^k \right) = \frac{\text{ord}\left( n \right)}{\gcd(k,\text{ord}\left( n \right))} }$$
Property of disjoint cycles - if all elements are unique:
$${(a \; a_n)(a \; a_{n-1})\cdots(a \; a_1) = (a \; a_1 \; a_2 \cdots a_n)}$$
Inverses:
$${(a \; a_1 \; a_2 \cdots a_n)^{-1} = (a \; a_n \; a_{n-1} \cdots a_1)}$$
Inverse of multiple cycles:
$${(a_1 a_2 \cdots a_n)^{-1} = a_n^{-1} a_{n-1}^{-1} \cdots a_1^{-1}}$$
This holds for any group so it holds for the Symmetric Group where \(a_k\) is a cycle.
Landau's Function:
Landau's function is just the maximum order of \(S_n\).
This value is just the maximum LCM of all the integer partitions of n.
A subset of a group must satisfy 4 conditions to be a Subgroup:
One-step Subgroup Test:
If a non-empty subset is closed under the group operation and inverses then it's a subgroup.
The Subgroup Lattice of a group is a partially ordered set (poset) that consists of all subgroups of the group.
The Centralizer is the set of all elements in G that commute with a.
$${C_G(a) = {g \in G | ga = ag} }$$
The Center is the set of all elements in G that commute with every element of the group.
$${Z(G) = {g \in G | gx = xg} \; \forall \; x \in G }$$
The Normalizer is the set of all elements that when conjugated they remain in the group.
$${N_G(H) = {g \in G | gHg^{-1} = H} }$$
A cyclic group is a group that can be generated by a single element by exponentiating or repeating the group operation.
$${ G = \{ g^n \mid n \in \mathbb{Z} \} }$$
Every cyclic group is abelian.
For a finite cyclic group it has \( \phi(n) \) generators.
Every subgroup of a cyclic group is also cyclic.
$${ \langle g^k \rangle = \langle g^{\gcd(n, k)} \rangle }$$
Where n is the order of the group.
Let G be a finite cyclic group of order n. Then:
gk where gcd(k, n) = n / d.These facts give a full classification of the subgroup structure of G and the orders of its elements.