First we define some terms:

## Algebraic Structure (Grouppoid)

A Algebraic Structure (Gruppoid) (G,op) is a set G together with a binary operation op, where op : G x G -> G . G has to be closed over by op (op cannot be used to leave G)

## Halfgroup

A Halfgroup is a Grouppoid where op is associative, that is: ((x op y) op z) = (x op (y op z))

## Monoid

A Monoid is a Halfgroup where a *Neutral Element* e exists so that (e op x) = (x op e) = x for all a ∈ G

## Group

A Group is a Monoid where any element x has (one) *Inverse Element* so that .

## Commutative Group

A Commotiative Group (Abelian Group) is a Group where op is always commutative.

## Power of Elements of the Group

The nth Power of x (where n is a whole number), written , is defined as:

Then, the laws of power holds:

((+) and (⋅) are operations on the whole numbers)

### Group Homomorphism

A map φ : G -> H between (the sets of) groups (G,⨯) and (H,%) is called Group Homomorphism iff ∀a∈G ∀b∈G:

### Group Isomorphism

A Group Isomorphism is a Group Homomorphism where φ is also a bijection. The Groups G and H then are called isomorph, written (G ~= H).

### Kernel

The Kernel of φ, written ker φ, where φ is a Group Homomorphism, is defined as:

(where is the neutral element of H).

## Ring

A Ring is an Algebraic Structure (G,+,⋅) over a set G with two binary operations (+) and (⋅), such that all of the following hold:

- (G,+) is a Commutative Group.
- (G,⋅) is (at least) a Halfgroup.
- Distributivity holds: and

## Commutative Ring

A Ring (G,+,⋅) that is commutative *with respect to (⋅)* is called Commutative Ring.

## Integrity Ring

A Commutative Ring with an element "1" without divisors of zero is called Integrity Ring. There, one can *cancel out* common factors (the common factors are called *units* (as opposed to unity!)) in equations.

## Field

A Commutative Ring with an element "1" (which is not the same as "0"), where each element except 0 is a unit (can cancel out any common factor except 0), is called Field.

## Lie Rotation Groups

- SO(n): Group of rotations in n real dimensions.
- SU(n): Group of rotations in n complex dimensions.
- SP(n): Group of rotations in n Quaternion dimensions.

## Subgroup

If G is a group, H is a subset of G and let g be an element of G, then:

H is a Subgroup if H=g⋅H⋅g^{-1} (H is preserved under conjugation).

## Conjugation

where is the inverse.

The conjugate is a measure of how much the operation commutes.

## Funny Symbols

- a H={ah: h∈H}: left coset of a.
- |G:H|: number of left cosets of H in G.
- ⋉ left semidirect product.
- ⋊ right semidirect product.
- ⋈ bicrossed product.
- orb(s): Orbit: set of elements that can be reached.
- stab(s): Stabilizer: set of elements that don't move s.