Group axioms

Task number: 2442

Show that alo reduced axioms
  • [(A)] \(\forall a,b,c\in G: (ab)c=a(bc)\)
  • [(E')] \(\exists e\in G\ \forall a\in G: ae=a\)
  • [(I')] \(\forall a\in G\ \exists b\in G: ab=e\)
define grup, i.e., that these provide also the "missing" rules \(ea=a\) and \(ba=e\).

  • Hint

    Show first from which side we can cancel common factors.

  • Hint

    Derive and then use an auxiliary fact \(aab=aba\).

  • Solution

    It is possible to cancel from right: \(ca=da\ \Longrightarrow\ c=ce=cab=dab=de=d\).

    (The cancelaton from left cannot be deduced in the same way, since we do not know yet tha \(c=ec\).)

    We now deduce auxiliary facts: \(e=ab=(ae)b=aabb\), but also \(e=ee=abab\), hence after cancelation from right: \(aab=aba\).

    Now follows: \(ea=aba=aab=ae=a\).

    Now the left cancelation could be derived: \(bc=bd \ \Longrightarrow\ c=ec=abc=abd=ed=d\).

    The last fact \(e=ab=ba\) we get by the left cancelation of \(a\) from the already derived fact \(aab=aba\).

Difficulty level: Hard task
Proving or derivation task
Solution require uncommon idea
Cs translation
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