[Axiom-developer] algebras <=> groups

[root]

Axiom arranges algebras based on categorical properties such as
associative, commutative, has one or multiple zeros, etc. The
same is true, it appears, in group theory.

It appears to me that there must be a group associated with every
algebra with an isomorphism (the group rules have exact analogs
in the algebra rules, the group elements have exact analogs in
the algebra and vice-versa). 

(a) Is this true?
(b) Can you point me at a reference that details this?

If we were to do group theory in Axiom it would be unbelievably
sweet to arrange the group category structure side by side with
the algebra category structure. 

That would allow the "lifting" trick (e.g. solving a matrix problem
by mapping each matrix to the group element, using a group theorem,
and then mapping back to the answer or vice-versa). It would also
immediately give us algorithmic power to answer group questions in
the algebra (e.g. showing a word is the identity by mapping it to
matrices, combining the results, and mapping back the identity 
matrix).

This identification of group theory with algebra would also solve
the problem I've been pestering you about (of the categorical ordering
of groups based on properties).

Tim