Anonymous wrote: > Olschok: > >> If I >>> remember correctly, an abelian group is a group that has an invertible >>table. >> >>What is an "invertible table" ? By the way, a group is abelian if the >>group operation is commutative. > > Right. So, if you set up a table with all the set members on the horizontal > and vertical, it'll form an invertible diagonal because xy = yx.
An "invertible diagonal" ? ( Are you inventing these on the fly? :-)
> So, a group has to be closed, ASSOCIATE, have an identity, > and have an inverse?
The last one to mean, "each group element has an inverse".
> Whereas an abelian group has to have all the above, plus be commutative, > right? It's been 6 months, at least.
Well, unless you burned all your notes from the last term, you can look it up if you do not feel sure.
> >>those axioms in (a) are those for an abelian group. >>In most linear algebra courses, the definition of an (abelian) group >>should at least be mentioned because >> >>- the axioms from (a) can be remembered easier. > > I agree. It's odd...we focused on abelian groups (and groups in general) in > algebraic structures. We never mentioned abelian groups in linear, > even though it's considered a continuing course from algebraic.
Bad course organization perhaps. The natural choice would be to ask your lecturer about the connections.
>[...] >>- the definition of a field also includes them for the field addition. > > Vector field?
No. A field in this context is a ring such that every nonzero element has a multiplicative inverse. That is where the scalars come from that operate on the vectorspace via scalar multiplication.
For example the real numbers form a field in this sense. Perhaps only vector spaces over the reals are considered in your current course; in this case the above concept was probably never introduced explicitely.