Talk:Klein four-group

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[edit] 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? ;)

[edit] 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.

[edit] 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)