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.
Read more about this topic: Elementary Group Theory
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