Group Axiom Ordering: Proving Associativity First

[bologna121121]

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.

 

[mathman]

The proofs for identity and inverse will usually use associative law.

 

[micromass]

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]

In the definition, the axiom about the identity element must be stated before the one about inverses, since the latter mentions the identity element.

Let's say that you want to prove that the set of integers with the standard addition operation is a group. It's definitely OK to prove that x+(-x)=-x+x=0 for all x before you prove that x+0=0+x=x for all x. However, if you do it in this order, it's not clear that the first step actually proves that this set and addition operation satisfy the axiom about inverses, until after you have performed the second step, which establishes that 0 is an identity element of this addition operation.

For this reason, I would recommend that you at least do those two in the standard order. Your proof would still be valid if you do these two steps in the "wrong" order, but it would be harder to understand.

 

[blue_raver22]

I understand the importance of following a logical and systematic approach when proving mathematical concepts. In the case of group axioms, it is crucial to prove associativity first because it is the foundation for the other axioms.

Associativity states that the order of operations does not matter when combining elements in a group. This property is essential for proving the existence of an identity element and inverse elements, as it ensures that the operations will always yield the same result.

If we were to prove the existence of an identity element or inverse elements before proving associativity, we would be assuming that the operations are associative, which could lead to errors or inconsistencies in our proof.

By proving associativity first, we establish a solid basis for the other axioms and can confidently proceed with our proof. This approach follows the principle of building from the ground up, ensuring that each step in our proof is sound and valid.

In summary, proving associativity first in group axiom ordering is a crucial step in ensuring the validity and consistency of our proofs. It provides a strong foundation for the other axioms and allows us to confidently establish a set as a group.