alternative definition of group
Let the non-empty set satisfy the following three conditions:
I. For every two elements , of there is a unique element of .
II. For every three elements , , of the equation holds.
III. For every two elements and of there exists at least one such element and at least one such element of that .
Then the set forms a group.
Proof. If and are arbitrary elements, then there are at least one such and such that and . There are also such and that and . Thus we have
i.e. there is a unique neutral element in . Moreover, for any element there is at least one couple , such that . We then see that
i.e. has a unique neutralizing element .
|Title||alternative definition of group|
|Date of creation||2013-03-22 15:07:58|
|Last modified on||2013-03-22 15:07:58|
|Last modified by||pahio (2872)|