groups

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Groups

What is a group?

A Group ⟨G,∘⟩ is a set equipped with a binary operation ∘:G∘G↦G such that:

1. (a∘b)∘c=a∘(b∘c)∀a,b,c∈G (associative property)
2. ∃e|a∘e=e∘a=a∀a∈G (neutral element)
3. ∃ā∀a∈G|a∘ā=ā∘a=e (inverse element)

example groups are: ⟨ℤ,+⟩,⟨ℚ+,⋅⟩,⟨Sₙ,∘⟩

Special properties of groups

• if a∘b=b∘a then the group is commutative(Abelian)

if ∘ is denoted +, then e is usually denoted 0 and ā as -a
if ∘ is denoted ⋅, then e is usually denoted 1 and ā as a⁻¹

Generator (DE:Erzeugendensystem)

given E⊆G for ⟨G,∘⟩, then E is a generator if and only if there exists a way to write any member of G as a product (∘) of some members of E and/or their inverses

Subgroup (DE:Untergruppe)

given U⊆G for ⟨G,∘⟩, then ⟨U,∘⟩ is a subgroup if

• ∀x,y∈U:x∘y∈U
• ∀x∈U:x⁻¹∈U