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These days, I'm interested in group theory. Why is the group axiom the way it is now? As an example, among mathematics, algebra has fundamental properties called associative property, commutative property, and distributive property. However, among these, only the associative property is included in the group axioms. And among the above properties, when the commutative property is established, it is called a special term, Abelian group. And there are certainly many good properties in the Abelian group. So I want to know why the group axiom ended up being the way it is now.

And I understand that the group theory comes from linear equations. So, in my personal opinion, I think that the generalization of possible solutions in linear equations is reflected in the group axiom, but I am asking this question because I want to know for sure. And as an additional question, what was the reason in the history of mathematics that led to thinking of groups based on linear equations?

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    $\begingroup$ As with most mathematics, Group theory doesn't come from any one field or problem. In the 19th century there were a lot of results that would later be systematically organized and streamlined into what we know today as Group Theory. Most notably, Galois was among the first mathematicians to work with groups in the modern sense, and his motivation wasn't linear equations per se but the problem of solvability of polynomials in radicals. $\endgroup$ Commented Feb 1 at 8:30
  • $\begingroup$ @RandyMarsh Thanks for your comment. Then, it seems that the connection with the linear equation was discovered later. $\endgroup$ Commented Feb 1 at 8:51
  • $\begingroup$ To incorporate the distributive property look at rings. Also, abelian (or commutative) groups are distinct from non-abelian. $\endgroup$
    – i can try
    Commented Feb 1 at 8:52
  • $\begingroup$ See also math.stackexchange.com/questions/388351/… $\endgroup$
    – lhf
    Commented Feb 1 at 12:54
  • $\begingroup$ You might consider asking your quaternion question as a separate post. $\endgroup$
    – Karl
    Commented Feb 1 at 17:36

3 Answers 3

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The group axioms reflect the properties of bijections from a set to itself (and more generally, of automorphisms in any category), with function composition as the group operation: composition is inherently associative, and we have identity and inverse functions. In fact, every group is isomorphic to an automorphism group.

Composition isn't generally commutative. For example, rotating a shape 90 degrees around the origin doesn't commute with reflecting across the $x$-axis, so the group generated by these two operations (which is a dihedral group with 8 elements) is not abelian.

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  • $\begingroup$ Hello thank you for your answer. Then, can we assume that galois, who was the first to study groups in the modern sense, also viewed groups as functions, so the group axiom has been maintained in that form until now? $\endgroup$ Commented Feb 1 at 9:11
  • $\begingroup$ I don't know the history, actually. I just meant to give the "hindsight" perspective on why the group axioms are a useful bundle: because they model "composable, invertible operations", which are ubiquitous in math. $\endgroup$
    – Karl
    Commented Feb 1 at 9:20
  • $\begingroup$ Galois’s Mémoire sur les conditions de résolubilité des équations par radicaux introduced the concept of a groupe de substitutions which we'd call permutations today, but the same underlying concept of composable, invertible, but not necessarily abelian. $\endgroup$ Commented Feb 1 at 12:35
  • $\begingroup$ I think @WarrenMoore is right. The abstract notion of "group" is more recent than Galois, who used only groups of permutations. Abstract groups are part of the innovation called modern algebra in the early 20th century. $\endgroup$
    – GEdgar
    Commented Feb 1 at 16:28
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I can recommend reading Hans Wussing, The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, see for example here.

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    $\begingroup$ This doesn't answer the question. $\endgroup$
    – James K
    Commented Feb 2 at 7:07
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    $\begingroup$ But it does, have you read the book, or at least the summary? As a student I read it in German (no English translation available at the time) and it perfectly takes the reader along the genesis of the concept of a group. $\endgroup$ Commented Feb 2 at 8:23
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    $\begingroup$ That's not the point. This is a meta answer. It doesn't answer the question. It just says where the answer can be found. That might be a useful addition to an answer. But as it stands this doesn't attempt to answer the question A person reading your answer would not know why group theory took its current form. This is a "link only answer" and so not useful. Hence my downvote. $\endgroup$
    – James K
    Commented Feb 2 at 23:15
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I really like Socratica's Treatment of Groups for a lot of these questions. But in general, we got to group theory because it was useful to get to group theory.

You mention 3 substantially important axioms: associativity, commutativity, and distributivity. Of these, distributivity requires 2 operations, so it's really handled separately with other abstract objects. In particular, vector spaces are distributive and that gives them a lot of power!

Associativity is an interesting beast. It appears to have a very special place in the heart of mathematics. Non-associativity does exist, but its either pushed to the side or commonly we will embed the non-associative algebra in something that is associative, and then analyze that. I've tried to inquire deeper, but it does seem to be something that is informally important to mathematicians.

We do explore commutative systems. But so far we have found more application for associative systems than commutative ones. Usually commutativity is layered on later, as we see with Abelian groups.

What I think gives groups their "fundamental" feel is that, for some reason, the 3 properties of groups permit incredible power. The fact that we can fully categorize the finite simple groups is pretty marvelous. And, of course, we simply find that group patterns show up in a lot of really useful problems. We could re-invent all of the principles every time, or we can start from proving what must be true for all groups, and go from there.

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