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I'm reading a simple introduction to groups. A group is said to be a set satisfying the followingaxioms(called the 'group axioms'):

1) Associativity.

2) There is a neutral element.

3) Every element has an inverse element.

4) Closure.

My questions is simply: why are they called axioms? I thought an axiom was something we take as a starting point,definingit to be true and then deduce something from it (possibly together with other axioms). Why are 1-4 not just the definition of a group?

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# Why 'axioms'?

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