Can there be more than one definition of a GROUP?

Paulibus

I'm reading a book about Group Theory (by Mario Livio: The Equation that Couldn't be Solved ). On page 46 he explains that four rules and one operation define a group: The rules are Closure, Associativity, the existence of an Identity Element and finally the existence of an Inverse. He cites all the integers (positive and negative) and zero as an example of a group; in this case with the single group operation being addition. A lot seems to depend on, and follow from, this simple definition, which nevertheless to me looks a bit arbitrary.

I know that numbers were invented a long time ago, perhaps in the Middle East to quantify resources like sheep and goats, or as labels for tally marks. I guess that positive counting integers handled this requirement, together with the two operations of addition and subtraction, variants on the actions make more and make less. Who knows or now cares? Negative integers and zero were postulated sometime later I think, as extra integers.

If this Group were instead defined as three rules (the first three I mentioned) and two operations (rather than one) ", i.e. do something (in this case addition) and do the opposite (here subtraction) between any pair of members, would this be an adequate definition of of the group?

And why couldn't one go further and manage with only two rules (the first two), but three operations, by including zero as the operation do nothing? Livio seeems to like this later in his book (Chapter Six).

Or would such flexibilty in definition cause trouble with other types of Groups?

Elroch

It is always possible to find variations on the definition of a class of algebraic objects (such as groups). In some cases such variations lead to a different class of algebraic objects, in others they are merely an alternative definition which defines the same class of algebraic objects.

Dropping the fourth group axioms defines a larger class of algebraic objects known as monoids. An example of a monoid that is not a group is the natural numbers {1,2,3,...} with multiplication as the operation. Another example is the non-negative integers {0,1,2,3 ...} with addition as the operation. Monoids are perfectly valid algebraic objects, but not everything that is true for all groups is true for all monoids.

But if you include a second operation which is the inverse of the first operation, you need an axiom to show that it is the inverse to multiplication ( eg (b * a)/a = b for all a and b in the group. With such an axiom you can ensure you are defining the same class of algebraic objects as you do with the usual group axioms.

When you have shown that two sets of definitions lead to the same class of objects, it doesn't matter which set you use - it's merely a matter of convenience.

[Note: remember that the operation in groups is generally not commutative]

Paulibus

Thanks for this helpful reply and the pointer to Wikipedia. I'd not heard of Mooids before.

Stephen Tashi

Rather than thinking of sets numbers as the typical example of a group, it would be better to think of monoids, groups etc. as exemplified by sets of functions. If you think of a set of functions and the "operation" as the compositions of functions, you will be reminded that the operation need not be comutative, f(g(x)) isn't always equal to g(f(x)). The functions most closely associated with finite groups are "permutations". ( It's better to think of permuations as functions than "ordered arrangements".)

Paulibus

I was being too focussed on a layman's definition of a group. Only excuse is: that's what I am, as far as mathematicians are concerned. Thanks. I'll be taking a broader view in future.