Elementary Group Theory - Alternative Axioms

Alternative Axioms

The pair of axioms A3 and A4 may be replaced either by the pair:

• A3’, left neutral. There exists an such that for all, .
• A4’, left inverse. For each, there exists an element such that .

or by the pair:

• A3”, right neutral. There exists an such that for all, .
• A4”, right inverse. For each, there exists an element such that .

These evidently weaker axiom pairs are trivial consequences of A3 and A4. We will now show that the nontrivial converse is also true. Given a left neutral element and for any given then A4’ says there exists an such that .

Theorem 1.2:

Proof. Let be an inverse of Then:

This establishes A4 (and hence A4”).

Theorem 1.2a:

Proof.

This establishes A3 (and hence A3”).

Theorem: Given A1 and A2, A3’ and A4’ imply A3 and A4.

Proof. Theorems 1.2 and 1.2a.

Theorem: Given A1 and A2, A3” and A4” imply A3 and A4.

Proof. Similar to the above.