Definition (left-sided)
A group $ \langle G, \cdot \rangle $ is a set G and a Binary Operation_Old Kiwi $ \cdot $ on G (closed over G by definition) such that the group axioms hold:
- Associativity: $ a\cdot(b\cdot c) = (a\cdot b)\cdot c $ $ \forall a,b,c \in G $
- Identity: $ \exists e\in G $ such that $ e\cdot a = a $ $ \forall a \in G $
- Inverse: $ \forall a\in G $ $ \exists a^{-1}\in G $ such that $ a^{-1}\cdot a = e $
Notation
Groups written additively use + to denote their Binary Operation_Old Kiwi, 0 to denote their identity, $ -a $ to denote the inverse of element $ a $, and $ na $ to denote $ a + a + \ldots + a $ ($ n $ terms).
Groups written multiplicatively use $ \cdot $ or juxtaposition to denote their Binary Operation_Old Kiwi, 1 to denote their identity, $ a^{-1} $ to denote the inverse of element $ a $, and $ a^n $ to denote $ a \cdot a \cdot \ldots \cdot a $ ($ n $ terms).
Theorems
Element commutes with inverse
$ Thm: \forall a\in G $ $ a\cdot a^{-1} = a^{-1}\cdot a = 1 $
$ Prf: Since a^{-1}\in G, its inverse (a^{-1})^{-1}\in G with (a^{-1})^{-1}\cdot a^{-1} = 1 by the inverse axiom. But also a^{-1}\cdot a = 1 $
