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?