if we are set out to see whether <R,*> is a group or not then a*0 remains undefined since '*' was defines only on R\{0} . so one can't be sure that <R,*> is a binary structure to begin with. So i would say that one can't conclude that <R,*> is a group if have * as defined in the question.
the author of the book should be publicly flogged for posing such an ill-formed question.
as posed, <R,*> is not a group, because * has not been defined on all of R. but this is probably not the question the author meant to pose.
he probably meant to ask if <R-{0},*> was a group.
You say e/|e| is the identity element since x cannot be 0 and thus e is defined...however, it is not defined uniquely. An identity element is always unique (if, say, x and y are identity elements then xy=x and xy=y, so x=y), so something is amiss.
Remember that e is a real number, so try and place it on the number line. e=...? This will help you to see what is amiss...