Definition
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[1ZF](Solved on 2022-11-15) A group is a set \(G\) equipped with a binary operation \(*\), that associates an element \(a*b∈ G\) to each pair \(a,b∈ G\), respecting these properties.
Associative property: for any given \(a, b, c∈ G\) we have \((a*b)*c=a*(b*c)\).
Existence of the neutral element: an element denoted by \(e\) such that \(a*e=e*a=a\).
Existence of the inverse: each element \(a∈ G\) is associated with an inverse element \(a'\), such that \( a*a'=a'*a=e\). The inverse of the element \(a\) is often denoted by \(a^{{-1}}\) (or \(-a\) if the group is commutative). 1
A group is said to be commutative (or abelian) if moreover \(a*b=b*a\) for each pair \(a,b∈ G\).