context $G$
definiendum $ \langle G,* \rangle \in \mathrm{it}$
inclusion $\langle G,* \rangle \in \mathrm{monoid}(G)$
let $e$
such that $\forall g.\, e*a=a*e=a$
range $g,g^{-1}\in G$
postulate $\forall g.\,\exists g^{-1}.\;(g*g^{-1}=g^{-1}*g=e)$

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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