# Is there any significance in the order in which group axioms are presented?

I know books have slightly different ways of presenting the axioms, but I think they tend to go like this: Group is a set with a law of composition with closure that satisfies the following properties (i) associativity, i.e. a(bc) = (ab)c (ii) identity, i.e. 1a = a1 = a (iii) inverse, i.e. every element a has an inverse such that ab = ba = 1.

Michael Artin's "Algebra" made no mention of this, but Fraleigh's "A First Course in Abstract Algebra" implied that it was important that the axioms were in that specific order, and wouldn't work any other way.

Why is this?

Thanks in advance.

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Talking about inverses does not make sense until you have introduced the identity, so you have to do (ii) before (iii). – Mikko Korhonen Aug 12 '12 at 11:45
@m.k. In the language of groups $(\cdot, e)$, you have a distinguished element $e$. You can define the inverse of $g$ to be an element $g^{-1}$ such that $gg^{-1} = e$. After saying this, you can state that this special element $e$ has the property that for all $g$, $ge = g$. – William Aug 12 '12 at 13:39

## 2 Answers

As m.k. says in the comments, in order for the axioms to even make sense (as they are usually presented) it is necessary to state identity before inverses. Other than that the axioms may be stated in any order one wishes. (As William says, it is possible to write down the identity $e$ as a distinguished symbol instead of to merely posit the existence of the identity as an axiom, and in that case it is not even necessary to do this.)

However, I think the order you've stated them in is a good one because, as Hurkyl says, taking an initial segment of the axioms leads to other theories of independent interest:

• Taking only associativity gives you a semigroup.
• Taking associativity and identity gives you a monoid.

Just as groups abstract collections of symmetries of a set, semigroups and monoids abstract collections of functions from a set to itself (not necessarily invertible), and for that reason they are also natural and important to study. However, it is often harder to say anything useful about them. For example, finite groups have some nice number-theoretic structure to them because of results like Lagrange's theorem, and Lagrange's theorem fails very badly for finite monoids.

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Also, the combinatorics are more complex: there are only 5 groups of order 8, but 1,843,120,128 semigroups of order 8 (satoh et al, 1994). – alancalvitti Aug 12 '12 at 15:21

The ordering of axioms has no technical content: it's purely an issue of exposition.

Sometimes, an initial segment of axioms will present a theory that is interesting in its own right, and you will want to study that before adding additional axioms. For example, the modern presentation of formal Euclidean geometry usually starts with the axioms of incidence geometry.

Sometimes, the ramifications of an axiom might make more sense if other axioms are presented first.

Sometimes, it is simply much easier to state an axiom in the presence of prior ones. For example, in Zermelo set theory, the axiom of infinity is much easier to state if you can make reference to the empty set, and so it is simpler to state it after the axiom of the empty set.

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