The Unapologetic Mathematician

Mathematics for the interested outsider


A good ramp up into abstract algebra is the idea of a group. Groups show up everywhere in mathematics, and getting a feel for working with them really helps you learn about other algebraic notions.

There are a number of ways to think about groups, but for now I’ll stick with a very concrete, hands-on approach. This is the sort of thing you’d run into in a first undergraduate course in abstract (or “modern”) algebra.

So, a group is basically a set (a collection of elements) with some notion of composition defined which satisfies certain rules. That is, given two elements a and b of a group, there’s a way to stick them together to give a new element ab of the group. Then there are the

    Axioms of Group Theory

  1. Composition is associative. That is, if we have three elements a, b, and c, the two elements (ab)c and a(bc) are equal.
  2. There is an identity. That is, there is an element (usually denoted e) so that ae=a=ea.
  3. Every element has an inverse. That is, for every element a there is another element a^{-1} so that aa^{-1}=e=a^{-1}a.

That’s all well and good, but if this is the first time thinking about an algebraic structure like this it doesn’t really tell you anything. What you need (after the jump) are a few

    Examples of groups

  • The integers with addition as the operation
  • The rational numbers with addition as the operation
  • The nonzero rational numbers with multiplication as the operation
  • The real numbers with addition as the operation
  • The nonzero real numbers with multiplication as the operation
  • The numbers on a clock face with addition “modulo 12” as the operation
  • Rearrangements of three distinct items on a line with composition of rearrangements as the operation
  • Rotations of three-dimensional space with composition of rotations as the operation

The first five examples come up a lot, and many other systems are based on them. It shouldn’t take much thought to verify the axioms for them.

The sixth example, “clock addition”, is an extremely important one. The term “modulo 12” could use some explanation, though. All this means is that when I add or subtract numbers I might get something outside the range of from one to twelve that actually show up on a clock. We handle this by just adding or subtracting twelves until we get back into that range. We do this all the time without thinking too much about it: if it’s 11:00 now and I want to go for tea at 4:00 I subtract 11 from 4 to get -7. This is below the proper range, so I add 12 to get 5 — tea is five hours away. It takes a bit more work to pick out the identity and inverses here, but it’s not too difficult. Also, note that there’s nothing really special about 12. We can work “modulo n” for any number n. We call this example Z12, and the general case Zn.

For the seventh example, imagine I have three objects — a, b, and c — and I want to line them up. There are six ways I can do it:
a b c
a c b
b a c
b c a
c a b
c b a

How do I get from the first arrangement to the third? I swap the first two objects. That transformation we call a “permutation” of the three objects. Going from the first to the fifth is another permutation by taking the object from the end and sticking it at the beginning of the line. Doing these two permutations one after the other I swap the first two objects, then take the third and move it to the front, taking (a\,b\,c) to (c\,b\,a). If I move the third object to the end first, though, the composition takes (a\,b\,c) to (a\,c\,b).

This illustrates an important point about groups: we never assumed that the operation is “commutative”. The order we do things matters in general. This isn’t familiar from arithmetic, but it’s common in everyday life. If I’m driving with my arm out the window, there’s a big difference between these two procedures

  • Pull my arm inside, then roll up the window
  • Roll up the window first, then pull my arm inside

The last example is also not commutative. It’s also got a new twist that’s also in the examples involving the real numbers: the group elements form a continuum. For integers with addition we’re just looking at this number or that number, and they’re nicely separated from each other. We can slide our way around the group of rotations from one rotation to another, which adds all sorts of new geometric structure to the group. This is an example of what we call a “Lie group”, after Sophus Lie (pronounced “lee”). Given how important they are I’m sure I’ll mention them more in the future.

I’ll close for now with a few basic statements. I’ll leave the proofs for interested readers. They aren’t too hard from the basic axioms of a group.

  • A group has only one identity element
  • An element of a group has only one inverse
  • For elements a and b of a group, we have (ab)^{-1}=b^{-1}a^{-1}
  • For an element a of a group, we have (a^{-1})^{-1}=a
  • For any two elements a and b of a group, the equations ax=b and xa=b have unique solutions in the group

January 30, 2007 - Posted by | Algebra, Group theory, Structure of Groups


  1. for the last thing in the “for the reader to prove” section, it seems that the solution for x in ax=b will not be the same solution for x in xa=b unless a^(-1) and b commute.

    Comment by andy | January 31, 2007 | Reply

  2. Correct: they dont have the same solutions. What’s to be shown is that the solution for each equation exists and is unique.

    Comment by John Armstrong | January 31, 2007 | Reply

  3. Interesting blog. Hope it grows in popularity.

    About the permutation group, it is perhaps better not to talk both about `swapping’ and `sticking it at the beginning’ as these two are different operations, the latter is made up of several swap operations. Going from { a b c } to { c a b } is two swaps, from { a b c } to { a c b } to { c a b }. Using only swaps is also useful in determining even and odd permutations of n objects (cyclic is odd for even n).

    Comment by Amitabha | January 31, 2007 | Reply

  4. You’re exactly right, Amitabha, that every permutation can be built up from transpositions. However, I’m trying to give examples of different kinds of permutations for an audience that may not be as accustomed to them. Rest assured that I will have to return to the transpositions picture when I eventually need to talk about the signum representation.

    Comment by John Armstrong | January 31, 2007 | Reply

  5. most people now start with set, then move up to groupoid, semi-group, then quasigroup, then loop, then monoid, then finally group

    Comment by babi | January 31, 2007 | Reply

  6. babi: If this were a formal exposition I would start with set theory, yes. This isn’t a formal exposition, though. I’m trying to give the idea of groups for a somewhat general audience.

    That said, I’m not sure what “most people” you’re talking about. It’s a lot easier to point to examples of groups than to examples of semigroups, so the notion of a group is easier to communicate at an introductory level. Almost all the basic abstract algebra texts I’ve seen — Hungerford, Judson, and Jacobson for examples — start in with groups. Even Lang’s Algebra only starts with monoids and then moves to groups within a couple pages.

    The only major departure I’ve seen is older texts like Birkhoff & MacLane, which start with integral domains to build off of a basic understanding of the integers. I have yet to see a single basic algebra text which even mentions loops, except possibly in passing. Can you provide an example of one? Bourbaki doesn’t count since I don’t know anyone who would seriously try to teach a first course in anything from Bourbaki.

    Anyhow, if I wanted to be thorough I’d have to throw in magmas, categories, n-categories, and so on. If I wanted to do that I’d start from categories, steal the exposition from Lawvere, and promptly confuse everyone. Groups are rather easy for a novice to pick up, while being useful enough to lead into quandles, which is what I really want to get to pretty soon.

    Comment by John Armstrong | January 31, 2007 | Reply

  7. Motivation *before* formalism, I always say. Nice introduction to group theory.

    Comment by Tom LaGatta | January 31, 2007 | Reply

  8. Just so that everybody’s on the same page: I’m pretty sure that what babi calls “groupoids” are the same thing as what John (in the next comment) calls “magmas”.

    Comment by Toby Bartels | January 31, 2007 | Reply

  9. Toby, you mean that some people use “groupoid” for a different thing? Then what do they call groupids?

    Comment by John Armstrong | January 31, 2007 | Reply

  10. Allow me to introduce a discordant note. While this is a well-written introduction to groups, I’m pretty sure virtually no one from “the general audience”, without prior experience with rigorous college-level math, will be able to really understand it.

    It may appear to you as though you’re starting from scratch with the basic definitions anyone can understand, but in fact to even follow your train of thought requires prior experience with these ideas, in no particular order:

    1. That a set is a collection of entities treated as having no structure, and what that means
    2. That an algebraic structure is generally a set + some operations defined axiomatically, and how that connects to familiar objects not previously so considered
    3. Why sticking a and b together is called “composition”, what does it mean that it’s another element of the group (it’s just a and b written together!).
    4. What’s ‘associative’ and why it’s important.
    5. What does it mean to have an operation that’s not given explicitly, but only constrained axiomatically, and how does one do things with it if you don’t know what it means.
    6. ugh, rational numbers… I remembered once what those are exactly… oh you mean fractions!
    7. How the hell did you get from permutations of letters to some mysterious “objects”?

    and so on and so forth.

    While I appreciate the spirit in which you’re approaching your new blog (which looks interesting, by the way, thanks), I think you’re vastly underestimating the difficulty of explaining modern math structures to general public. I tried to further speculate on the nature of such underestimating in a post inspired by your effort.

    Comment by Anatoly | February 1, 2007 | Reply

  11. Thanks for the comments, Anatoly. You raise some interesting points, and I’ll try to keep them in mind as I continue these basic-concepts posts. At the outset, you’re right that what I’m trying to pull off here is difficult. You’re also dead-on that I have a long row to hoe to model an average reader’s thought processes, especially since I diverge from the norm you point out about learning abstract algebra only after basic linear algebra.

    I’m not quite as pessimistic, though. If I were trying to teach people to actually use group theory I’d be with you all the way. Instead, I’m trying to give people some of the flavor of these subjects with as few outright lies as possible. This game is downright popular in theoretical physicis, with such august authors as Greene, Hawking, and Penrose. In fact, Penrose’s The Road to Reality is a great inspiration to my project.

    I’ve actually had some pretty good results using this sort of approach in the past, albeit face-to-face. Can my friends past whom I’ve thrown some ideas actually work in group theory, or even read a current paper in the subject? Probably not. Do they have an idea of what group theorists do? Definitely more than before we talked.

    I’m also hoping that using many examples will help clarify the abstract axioms I’m laying down. I’m very explicitly trying not to be another Bourbaki here.

    My next post in this series will likely be an attempt to clean up some of these questions you mention. Any other points I’m less than clear on in the future, please bring them up and I’ll be glad to go back and run over them again.

    Comment by John Armstrong | February 1, 2007 | Reply

  12. has recently added some simple group theory. Since they don’t have a permutation notation, or even a plan for decimal numbers larger than 9, it may be a while before you can use them for introductions to abstract algebra. I found the following related proofs (if you are comfortable with the “metalogic” used by the author).

    A group has only one identity element

    An element of a group has only one inverse

    For elements a and b of a group, we have (ab)^-1=b^-1 a^-1

    For an element a of a group, we have (a^-1)^-1=a

    For any two elements a and b of a group, the equations ax=b and xa=b have unique solutions in the group
    Aha. This would seem to be some low-hanging fruit. Now if I could only figure out how to enter proofs into their system.

    Comment by R Penner | February 3, 2007 | Reply

  13. As I understand it, MetaMath is an attempt to actually present formal proofs of mathematical theorems? It’s interesting, but I think that even the notion of a formal system and a formal proof would go over the head of most basic readers. It’s going in the opposite direction from Bourbaki that I want to take this discussion.

    Thanks for the heads-up that they’re doing this sort of thing now, though.

    Comment by John Armstrong | February 3, 2007 | Reply

  14. […] few more groups I want to throw out a few more examples of groups before I move deeper into the […]

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  16. Though six months late, as a non-mathematician I understand what John has written. I think there are many semi-techies on the net that are:

    1. Interested in whats going on in math and science
    2. Have some training in them

    I work in publishing .

    Comment by Michael D. Cassidy | July 12, 2007 | Reply

  17. Michael’s remark above seems entirely correct to me.

    And, I think a good reason for starting with groups rather than say monoids or semigroups is that groups have some fairly spectacular but accessible results, such as Lagrange’s theorem, whereas I’m not aware of a single surprising fact about semigroups that you could explain to to somebody who knew only the semigroup axiom and sixth grade arithmetic, and for monoids there’s nothing but the uniqueness of the identity, which might as well be presented in the intro to groups.

    Comment by MathOutsider | October 13, 2007 | Reply

    • For every finite semigroup F which has an identity and an inverse, the order of every subsemigroup S, with an identity and inverse both identical to those of F, of F divides the order of S. Or more generally, for any “surprising fact” about groups and subgroups, you can just restate it as a surprising fact about semigroups which have an identity and inverse and subsemigroups which have the same identity and inverse as the semigroup they are subsets of. After all, a group can get defined “merely” as a semigroup which also has an identity and an inverse. For monoids, you can do something similar also.

      Additionally, the uniqueness of the identity element for a set with some binary opeartion holds even in the abscence of *any* axiom.

      I also don’t know if you’ll find this sueprising or not, but say we measure the length of expressions by the number of symbols they have (parentheses get counted). For any semigroup expression E written in infix notation you can move parentheses around as you like obtaining E’, and for all E, E’, E=E’, no matter what lengths E and E’ have. For semigroup expressions written in prefix notation (and infix and postfix notation also), no matter what length two given expression have, as long two expressions having the same length come as meaningful expressions, and have the variables appear in the same order, they will come as equal expressions. For instance, for all semigroups with binary operation S, I know SSSabcd=SSabScd just because they both come as meangingful, all variables appear in the same order, and they have the same length. Or perhaps better…


      Comment by Doug Spoonwood | May 16, 2011 | Reply

  18. […] we know what a group is and what a subgroup is. Today I want to talk about the cosets of a subgroup in a group […]

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