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## Group TheoryFirst 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) ## HalfgroupA Halfgroup is a Grouppoid where op is associative, that is: ((x op y) op z) = (x op (y op z)) ## MonoidA Monoid is a Halfgroup where a ## GroupA Group is a Monoid where any element x has (one) ## Commutative GroupA Commotiative Group (Abelian Group) is a Group where op is always commutative. ## Power of Elements of the GroupThe 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 HomomorphismA map φ : G -> H between (the sets of) groups (G,⨯) and (H,%) is called Group Homomorphism iff ∀a∈G ∀b∈G: ## Group IsomorphismA Group Isomorphism is a Group Homomorphism where φ is also a bijection. The Groups G and H then are called isomorph, written (G ~= H). ## KernelThe Kernel of φ, written ker φ, where φ is a Group Homomorphism, is defined as: (where is the neutral element of H). ## RingA 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 RingA Ring (G,+,⋅) that is commutative ## Integrity RingA Commutative Ring with an element "1" without divisors of zero is called Integrity Ring. There, one can ## FieldA 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.
## SubgroupIf 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). ## Conjugationwhere 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.
Author: Danny (remove the ".nospam" to send) Last modification on: Sat, 04 May 2024 . |