# Talk:Klein four-group

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Field: Algebra

## multiplication table

I hate this not-working links... I'm not a mathematician but isn't a Cayley table essentially a multiplication table? Can't we just say

`Its [[multiplication table|Cayley table]] is given by:`

plz? ;)

I am a mathematician (though in a different subfield), and I second this. Similarly, I'd much prefer use of the term direct product rather than a direct sum, since the former has a much clearer definition for my students (like a Cartesian product), while a direct sum is only known by graduate students. Also, here's an example figure with right symmetry group.

## Name

It is not Viergruppe but Vierergruppe (fours group). I have discussed this with some German mathematicians and it seems that the attribution "Klein" originated as a joke. There is the question whether the Klein group is the abstract group or not. Consensus suggests it is.

The attribution "Klein" is not a joke. It was Felix Klein who coined the name Vierergruppe.

## Automorphism Group of what Graph?

I don't think the Klein four group is the automophism group of any graph, but it certainly isn't the one of this graph: One can exchange the two related vertices alone, so this is a automorphism and one can exchange the two vertices that are not related to any other, this is another automorphism.

...and one can do both, or neither, giving two more automorphisms. Thus we have four automorphisms which commute exactly like the table says they should — looks like the Vierergruppe to me. Or am I misunderstanding something? —Ilmari Karonen (talk) 23:27, 29 November 2005 (UTC)
Looks more like isomorphism to me: The permutations aren't the same, but the groups are isomorphic. There actually is no graph which automorphism group is exactly the Vierergruppe. (Or am /I/ missing something? -- Does the article mean equal or isomorphic?)
Well, according to the article the name Klein four-group applies to Z2×Z2 or to any group isomorphic to it. So I'd say the concept of the Vierergruppe is effectively only defined up to isomorphism. Of course, that's a matter of definition — one could always pick one specific group out of the equivalence class and call it the canonical Vierergruppe — but the article doesn't appear to do so. —Ilmari Karonen (talk) 15:14, 30 November 2005 (UTC)

### Solution

The Klein four-group is not the automorphism group of any simple graph (this is apparently a basic theorem). A credible proof is given on PlanetMath. The graph in this article was simple (having at most one edge joining each pair of nodes), and therefore wrong. I have removed it.

This was reverted with a correction EmmetCaulfield 13:35, 21 October 2007 (UTC)

The Klein 4-group is the automorphism group of the following nonsimple graph:

```     ________
/        \
O          O
\________/

```

Having said that, I don't think it's very useful to help one picture the group to say what graph it's an automorphism group of. Much more useful, IMVHO, would be to show that V4 is a symmetry group with a diamond-like figure with a non-unity aspect-ratio, like this:

```·····························
··············Y··············
··············|··············
···X----------+----------X···
··············|··············
··············Y··············
·····························
```

Which has the following symmetries only: 'x' (reflection in x), 'y' (reflection in y), and 'r' (180 degree rotation). Together with the identity (rotation through zero, if you like), we have:

``` V| 1 x y r
--+---------
1| 1 x y r
x| x 1 r y
y| y r 1 x
r| r y x 1
```

Which is clearly V4 (up to isomorphism). That helps me understand it much better. EmmetCaulfield 16:20, 19 October 2007 (UTC)

I think it is nice to understand small groups as automorphism groups of pictures.

It would be a good addition to include *first* a simple picture of a rectangle and indicate that its automorphism group is the Klein four group as a subgroup of A4. Certainly it is a better example in the abstract, but the simple graph example is also nice and already has a picture, so it should stick around too.

If it helps: An automorphism of a graph is a permutation of the vertices (which we can label with numbers 1..n) that happens to take edges to edges. List the edges as unordered sets of two vertices. In the example the set of edges is just { {1,2} }. Now we want to check each of the 4! = 24 elements of Sym(4), for whether they take {1,2} to {1,2} or not. You can just write them all down and check, but you can also reason that {3,4} has to be taken to {3,4}, and so you can only have elements of order dividing two: (), (1,2), (3,4), and (1,2)(3,4).

I think there are advantages to just talking about stabilizers of sets in "standard" actions of permutations on set like things, where the calculations would be more mechanical, but I think for an intro article a picture is nice.

You made some comments below about ways in which the article is crummy, that may basically come down to the article is hard to read and has odd facts lumped together. Rather than worry about whether the facts themselves are crummy, it might be easy to fix up the article just by adding some section headers. If the new reader sees a hundred examples all under == Huge list of examples == then he will know it's fine to look at the pictures and find one he thinks is pretty, and understand how the Klein four group is related to the picture.

At any rate, I think the article has potential, but it needs a few informative pictures (maybe check the Dihedral group article), and it needs to be organized so the reader can take just a single nibble from the elephantine sandwich which is the Klein four group. JackSchmidt 15:48, 21 October 2007 (UTC)

### Comment

I think there's a problem with this article, and at least some of the other ones on the dihedral and symmetric groups, in that the emphasis seems to be on abstruse demonstrations of the cleverness of the "owners" of the article rather than on the explanatory or illustrative power of an explanation. The "simple graph" here is a case in point: it may well be true that V4 is the automorphism group of the given graph with the given labeling of the vertices. Quite honestly, at my beginner's level of knowledge, it's very hard to tell without a great deal of mental gymnastics, and it seems clear from the above comments that I am not alone. What is certainly true is that V4 is not the automorphism group of any graph with the usual simple vertex labeling, but is the automorphism group of a doubly-connected two-node graph with this usual labeling, and that these facts have substantial explanatory value, since the automorphisms can be applied mentally to the graph without having to simultaneously juggle the labeling of vertices. Can we not teach the triple somersault first, and work in the back-flips later?

In a similar way, the article on D3/D6 (depending on your notation) explains it in terms of permutations. This is more of an explanation of S3, though, isn't it? Now, I know that the two are isomorphic, but surely it would be of much greater pedagogic value to explain D3 purely in terms of reflections and rotations, as the dihedral groups are defined, and then proceed to point out the isomorphism with S3, defined in terms of permutations? As it stands, the isomorphism is tacitly assumed, and the geometric interpretation is entirely ignored. But, precisely because of this isomorphism, the only "difference" is one of the semantics of the interpretation: if you interpret the group as permutations, you have S3, if you interpret it as reflection and rotation, you have D3. That these two are, or should be, the same, is not obvious unless you have substantial expertise, and those with this expertise are not the target audience. I am the target audience, and I'm confused.

Quite honestly, and I'm sorry if this sounds harsh, but I think these articles are pretty crummy. They appear to push abstruse cleverness to the front and push clarity to the rear or, worse, exclude it altogether. Apart from being unhelpful and inappropriate to an encyclopædia, it unnecessarily obscures that part of the transcendent beauty, wonder, unity, and intellectual achievement of group theory that would be accessible to people like me if only the explanation was better. That's a damn shame. It does no favours to mathematics. It does, however, make me appreciate the fact that my abstract algebra lecturer is not of the mindset which demands that one should be able to eat the elephant all in one go rather than one bite at a time.

I'm not seeking to attack anyone, I do think it's possible to be so accustomed to something that you honestly don't see how other people find it abstruse and I am often guilty of this myself in other areas. But while I am working on a clearer exposition for consideration here, I'd like the major contributors to this article to consider my comments, as they are intended, in a spirit of good faith and reflect on the expository power of the "up-front", emphasised content of these articles. EmmetCaulfield 13:35, 21 October 2007 (UTC)

Emmet, you are exactly right. This is a common problem with Wikipedia mathematics pages. The aim sometimes appears to be for the editors of the pages to show off their extensive knowledge of the most abstract areas of mathematics, rather than to explain the subject simply and clearly. —Preceding unsigned comment added by 128.243.220.42 (talk) 11:10, 3 October 2008 (UTC)

## The other group

There is a math music group called "The Klein Four Group", is there any article on them? (btw, I know this might be considered advertisement, but check out "Finite Simple Group (of Order 2)" on Youtube, it's pretty funny). Peter Stalin 02:22, 2 April 2007 (UTC)

We used to have an article on that music group which has rightly been deleted. It is non-encyclopedic (as you appear to have realized yourself). 141.35.26.61 20:57, 3 April 2007 (UTC)
Ah, yeah, thought as much...Peter Stalin 03:42, 5 April 2007 (UTC)
I'm kind of surprised that the music group doesn't have an entry, actually, given their infamy in the mathematics community - is the discussion pertaining to the deletion still available anywhere? If people are asking for information about a particular topic, I feel like that alone would indicate encyclopedic merit! Frankg 00:26, 10 September 2007 (UTC)

They have a CD now! I think they're notable simply because they sing about maths. Come on. Just because the first article was blatant advertisement doesn't mean a decent article can't be written.124.169.17.213 15:01, 7 November 2007 (UTC)

Amateur Transplants, a duo of singing medics, have their own page, so why not these guys?--87.194.86.28 (talk) 15:06, 18 November 2007 (UTC)