# Need the name of an abelian group

Discussion in 'Undergraduate Math' started by joostef, May 14, 2006.

1. ### joostefGuest

I've seen the following group presented in a few textbooks, mostly as
an exercise, but no special name or symbol has ever been attached to
it. Can anyone please tell me if it has a special designation?

G is an abelian group defined by the set of real numbers without -1,
under the operation a*b=a+ab+b, for any a,b in G.

joostef, May 14, 2006

2. ### Dave L. RenfroGuest

A good reference for something like this is

Frank J. Budden, "The Fascination of Groups",
Cambridge University Press, 1972.

I don't have a copy of Budden's book, but it's
an extremely leisurely treatment of elementary
group theory that has a large and diverse array
of group and group-like examples peppered throughout.

I don't know if your example has a special name, but
it is a special case of a more general construction:

Let (G,*) be a group and f: G --> G be a bijective
function (1-1 and onto), and let g be the inverse of f.
Then (G,*') is also a group, where the operation *' is
defined by

a *' b = g( f(a) * f(b) ).

Your group arises by taking (G,*) to be the
multiplicative group of nonzero real numbers
and f(x) = x + 1.

Below are three nice papers that illustrate the wide
variety of examples that can be obtained from this
general construction. (None of the later papers mention
any of the earlier papers, by the way.)

Robert A. Rosenbaum, "Some simple examples of groups",
American Mathematical Monthly 66 #12 (December 1959),
902-905.

Dan Kalman and P. Turner, "Algebraic structures with
exotic structures", International Journal of Mathematical
Education in Science and Technology 10 #2 (1979), 173-174.

Michael A. Carchidi, "Generating exotic-looking vector
spaces", College Mathematics Journal 29 #4 (September 1998),
304-308.

Dave L. Renfro

Dave L. Renfro, May 14, 2006

3. ### Ken PledgerGuest

Standard names for groups usually don't discriminate between
isomorphic copies. For example, C_3 is a general notation for a
cyclic group of order 3, of which there are infinitely many isomorphic
examples.

You probably know that the mapping x -> x + 1 is an
isomorphism from your group onto (R - {0}, x}, so yours is just the
real multiplicative group in disguise.

Ken Pledger.

Ken Pledger, May 15, 2006
4. ### FJGuest

Thanks Dave and Ken. I didn't make the connection that mine and the
multiplicative group of reals were isomorphic. Now I see this group is
not really anything too special.

Also, thanks for the references. I'll check them out next time I'm
near a university library.

-Francois Jooste.

FJ, May 15, 2006