I am soon going to explain to my students what the alternating group is. The definition is subtle.... one must prove that the notation of an "even permutation" is well defined.

There seem to be two proofs of this. One writes down a polynomial (a square root of an appropriately defined discriminant) for which the alternating group preserves the sign. The other is a fairly straightforward induction argument.

I can't think of any way of describing the alternating group that avoids giving the impression that it is something complicated and confusing. (And Gallian, Beachy and Blair, Saracino, Herstein etc. don't seem to offer too much beyond what I already described.) Is there something simple I can say?

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    $\begingroup$ Maybe let the permutation act on the standard basis of $\mathbb{R}^n$, and check the determinant of the corresponding linear map? Depends on how much linear algebra they have seen. $\endgroup$ – Steven Gubkin Oct 3 '14 at 1:23
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    $\begingroup$ Why are you presenting the alternating group in the first place? It is a nice example of a simple group, but so is $PSL_{\,n}(F)$. $\endgroup$ – user173 Oct 3 '14 at 1:24
  • $\begingroup$ @MattF. It is a standard topic, and I think that it would be impossible to pass most qualifying exams in group theory without knowing the definition. $\endgroup$ – Steven Gubkin Oct 3 '14 at 2:23
  • $\begingroup$ Am I missing something here? The fact that every element of $S_n$ can be written as a product of (adjacent) transpositions is very basic. $A_n$ is simply the group of elements which can be written as a product of an even number of transpositions. $A_n$ is easily checked to be a subgroup (by explicitly writing down the inverse/product), a single transposition is easily checked to not be in $A_n$, but for any two elements not in $A_n$ they're product is in $A_n$ so $A_n$ is an index 2 subgroup from Lagrange. You can define sign as the property of being in $A_n$ or not. $\endgroup$ – PVAL Oct 3 '14 at 4:57
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    $\begingroup$ @PVAL How do you prove that a single transposition is not in $A_n$? $\endgroup$ – Dag Oskar Madsen Oct 3 '14 at 12:26

It seems to me that the first thing to do is to define the signature. It is a very important morphism, and you need to have a bunch of example of morphisms to present to your students anyway. Only then is the alternating group really relevant, and you get all properties you want from what you will have done with the signature.

Of course, I am telling you to pass all the difficulties from the introduction of the alternating group to the introduction of the signature. But it seems that this way is much more natural, hence should be easier to convey convincingly.

For example, following the comment of PVAL, you can first show the important property that every permutation is a product transpositions, and ask whether the needed number of transposition is well defined. Of course it is not, point this out. Then what one could try is to associate to each permutation the minimal number of transposition needed to write it, but then it easily turns out that this map does not define a morphism (but it is linked to a lot of interesting math, from geometry of groups to work of Helfgott and others on the diameter of finite groups far arbitrary set of generators). Then, when you show the students that looking at the parity of the number of transpositions generating an element yields a morphism, you are in a very meaningful context.

There are other ways to introduce signature, for example looking at the number of pairs $(i,j)$ with $i<j$ whose order is changed by the permutation. The above one seems best to me right now, but if anyone can suggest other in comments I would be happy to have more choice when I have to teach that myself.

  • $\begingroup$ Interesting thoughts, thanks! $\endgroup$ – Frank Thorne Oct 3 '14 at 21:07

I basically agree with Benoit Kloeckner's answer. If you wanted to give a bit more context to the motivation, you could assign a sequence of exercises like the following:

Let $S_4$ act on $k[w, x,y,z]$ by permuting the variables.

  • What is the stabilizer of $w$? How big is the orbit of $w$?

  • What is the stabilizer of $w+x+2y+3z$? How big is the orbit of $w+x+2y+3z$?

  • What is the stabilizer of $w+x+2y+2z$? How big is the orbit of $w+x+2y+2z$?

  • What is the stabilizer of $wx+yz$? How big is the orbit of $wx+yz$?

Then, in class, start looking at the orbit of $(w-x)(w-y)(w-z)(x-y)(x-z)(y-z)$.

Any intro group theory course has to do a sequence of basic exercises on group actions and orbits, and this isn't worse than any other. Plus, it will be excellent preparation if any of them take Galois theory!

  • $\begingroup$ Thanks. This is really great! I will definitely keep this is mind the next time I teach group theory. $\endgroup$ – Steven Gubkin Oct 10 '14 at 15:52
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    $\begingroup$ As many people seem to use mostly algebraic examples like this (very good) one to teach group actions, I'd like to mention that Geometric examples should not be overlooked. In particular, $S_4$ is the group of direct isometries of the cube. This can be seen by looking at the way isometries permutes the largest diagonals. The morphism to $S_2$, aka signature, can also be seen easily geometrically: just look at the way isometries permutes the two regular tetrahedra inscribed in the cube (pictures). $\endgroup$ – Benoît Kloeckner Oct 11 '14 at 18:32

I'd second @BenoitKloeckner's suggestions about the larger discussion... a sort of dialectic, explaining why, although it's almost obvious that parity-of-number-of-2-cycles is well-defined, this is in fact the crux of the matter.

As @BenoitKloeckner says, indeed, the (minimal) length (in terms of the adjacent-transposition generators for $S_n$) is the number of pairs $i<j$ whose order is reversed by the permutation. It may be a little unclear how to see that this behaves as hoped under composition. One handy book-keeping device is (as Steve Gubkin noted) to let $S_n$ act by permutations on the coordinates of $\mathbb R^n$ (or any characteristic-zero field, but $\mathbb R$ is more familiar) and take sign-of-determinant.

Edit: also, a sort of shadow of the previous is to let $S_n$ permute the variables in a Vandermonde determinant... to define "sgn".

Indeed, the work to prove that determinants exist is a superset of the work to prove that the sgn character of $S_n$ exists. Some people would say that the $S_n$ case is the "field with one element" case of the other. :)


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