===== Group =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $G$ |
| @#FFBB00: definiendum | @#FFBB00: $ \langle G,* \rangle \in \mathrm{it}$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\langle G,* \rangle \in \mathrm{monoid}(G)$ |
| @#AADDEE: let         | @#AADDEE: $e$  |
| @#AADDEE: such that   | @#AADDEE: $\forall g.\, e*a=a*e=a$  |
| @#DDDDDD: range       | @#DDDDDD: $g,g^{-1}\in G$ |
| @#55EE55: postulate   | @#55EE55: $\forall g.\,\exists g^{-1}.\;(g*g^{-1}=g^{-1}*g=e)$ |

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=== Alternative definitions ===
== Sharper definitions ==
We could just define left units and left inverses and prove from the group axioms that they are already units and inverses.

== Group axioms explicitly in the first order language  ==
Let $\langle G,* \rangle $ be a set $G$ with a binary operation.

1. $\forall (a,b\in G).\ (a*b\in G)$

2. $\forall (a,b,c\in G).\ ((a*b)*c=a*(b*c))$

3. $\exists (e\in G).\ \forall (a\in G).\ (a*e=e*a=a) $

4. $\forall (a\in G).\ \exists (a^{-1}\in G).\ (a*a^{-1}=a^{-1}*a=e)$

The first axiom is already implied if "$*$" is a binary operation $*:G\times G\to G$.

For given $G$, the set $\text{group}(G)$ is the set of all pairs $\langle G,* \rangle$, containing $G$ itself, as well a binary operation which fulfills the group axioms. One generally calls $G$ the group, i.e. the set with respect to which the operation "$*$" is defined. 

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=== Subset of ===
[[Monoid]], [[Loop]]