# Group Axiom Ordering

## Main Question or Discussion Point

Hello,

In my abstract algebra class, my teacher really stresses that when you show that a set is a group by satisfying the axioms of a group (law of combination, associativity, identity element, inverse elements) these axioms MUST be proved in order.

This makes some amount of sense to me, as some axoims use other axioms in their definitions, but why must associativity be proved before the existence of the identity element or inverses? Thank you.

Related Linear and Abstract Algebra News on Phys.org
mathman
The proofs for identity and inverse will usually use associative law.

This makes some amount of sense to me, as some axoims use other axioms in their definitions, but why must associativity be proved before the existence of the identity element or inverses? Thank you.
You don't need to prove them in order. It's perfectly ok to show the existence of an identity element before associativity (just be sure that you never use associativity anywhere, but that's usually not the case).
It's a mystery to me why your teacher wants you to prove them in order.

Fredrik
Staff Emeritus