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单词 AlternativeDefinitionOfGroup
释义

alternative definition of group


The below theorem gives three conditions that form alternative postulatesMathworldPlanetmath.  It is not hard to show that they hold in the group defined ordinarily.

Theorem.

Let the non-empty set G satisfy the following three conditions:
I.     For every two elements a, b of G there is a unique element ab of G.
II.   For every three elements a, b, c of G the equation  (ab)c=a(bc)  holds.
III. For every two elements a and b of G there exists at least one such element x and at least one such element y of G that  xa=ay=b.
Then the set G forms a group.

Proof.  If a and b are arbitrary elements, then there are at least one such ea and such eb that  eaa=a  and  beb=b.  There are also such x and y that  xb=ea  and  ay=eb.  Thus we have

ea=xb=x(beb)=(xb)eb=eaeb=ea(ay)=(eaa)y=ay=eb,

i.e. there is a unique neutral elementPlanetmathPlanetmath e in G.  Moreover, for any element a there is at least one couple a, a′′ such that  aa=aa′′=e.  We then see that

a=ae=a(aa′′)=(aa)a′′=ea′′=a′′,

i.e. a has a unique neutralizing element a.

Titlealternative definition of group
Canonical nameAlternativeDefinitionOfGroup
Date of creation2013-03-22 15:07:58
Last modified on2013-03-22 15:07:58
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id13
Authorpahio (2872)
Entry typeTheorem
Classificationmsc 20A05
Classificationmsc 20-00
Classificationmsc 08A99
Related topicCharacterization
Related topicACharacterizationOfGroups
Related topicDivisionInGroup
Related topicMoreOnDivisionInGroups
Related topicLoopAndQuasigroup
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