Group Axioms, Vector Space Axioms, Inner Product Axioms.

Question

This might be a slightly silly question, apologies for that.

Anyways, I have been studying Linear Algebra and some Group Theory at University over the last few months and I started wondering why Axioms like the group axioms, vector space axioms etc are axioms, rather than just conditions that mathematical objects satisfy.

The reason I ask this is because the Group axioms especially, seem to me no different, in type, from say the conditions which a bijective function satisfies or the conditions an odd number satisfies.

So my main question is , why are group axioms, vector space axiom, inner product axioms, etc called axioms rather than simply conditions.

Any ideas will be appreciated.

Answer 1

While there is no hard-and-fast rule, here is a possible distinction between the use of axioms and conditions. The term axioms tends to be used when defining a very general object or theory, so that the vector space axioms define what it means to be a vector space. Once the general theory has been outlined, special objects are identified by satisfying certain conditions or properties. Thus we might first give group axioms and then later specify the conditions for a group to be abelian. I suppose you could say that axioms are foundational and set the context, whereas conditions are the features some subsequent objects of study might or might not enjoy.