# Meaning of “Komplex” (mathematical context)

I am currently struggling to understand the article “Zur Theorie der faktorisierbaren Gruppen”, by L. Redei in Acta Math. Acad. Sci. Hung. 1, 74--98 (1950).

As might be expected, there are several specialized German mathematical words that an ordinary dictionary fails to contain.

In particular I'm struggling with the word “Komplex”. In following the paragraph introducing the word “Komplex”, followed by my dubious translation.

Original:

Bezeichne H eine beliebig vorgelegte Gruppe. Für gewöhnlich versteht man unter einem Komplex K (von H) eine Teilmenge von H, oft kommt es aber auch vor, daß man dabei den Elementen des Komplexes eine Multiplizität zukommen läßt. Das wollen wir tun, so daß einen Komplex in der Form K=a1γ1,a2γ2, … annehmen, wobei γ123…, alle verschiedenen Elementen von H sind und jedes ai gleich 0,1,2 …∞ ist.

My translation:

Let H be any finite group. What we mean by a complex K (of H) is usually a subset of H, but where we allow the elements of the complex to have multiplicities. So we will write a complex in the form K=a1γ1,a2γ2, …, where γ123… are all different elements of H and each ai is one of 0,1,2 …∞.

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I think you should ask in the chat of Mathematica.SE for German-speaking Mathematicians helping you here. – Speravir Nov 22 '12 at 20:40
I think you mean chat.stackexchange.com/rooms/36 - this is a math question, not a Mathematica question. – Johannes Kloos Nov 25 '12 at 12:57
"... eine beliebig vergelegte Gruppe" is wrong. I think it's a typo but not sure what it should be. vorgelegt, perhaps? – Em1 Nov 26 '12 at 13:25
@Em1 : about "vergelegte" it’s a messy photocopy and I hesitated between "vOrgelegte" and "vErgelegte". In any case, I did not find it in the dictionary and my translation is pure guesswork – Ewan Delanoy Nov 26 '12 at 13:41
Ewan, I see, you cannot go to the chat. You must register first, I guess. And for sure you need a reputation score of 20 … sad that. I’m not a mathematician, so I cannot help here, but a special math dictionary should help you: google.com/… – Speravir Nov 26 '12 at 18:23

Here Komplex used in the meaning of group or cluster. Komplex can be used as meaning to be complex or as being a group of things which by virtue of being a group is complex. So group, collection or cluster of things.

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The definition does not seem to match chain complexes. If you take a look at the year of publication, then I suggest that the terminology of this author has not risen to prominence since then (while I am not a group theorist, so in a (tiny) subarea of group theory it may have survived). So I suppose there is no harm if you choose a term on your own (like complex). Maybe it is even more appropiate to choose new term, because complex is already overloaded with complex numbers, complex multiplication and chain complexes.

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Another guess, which may be completely wrong at all: en.wikipedia.org/wiki/Word_%28group_theory%29 – shuhalo Nov 30 '12 at 20:41

If I am not mistaken, `Komplex` is the short form for Kettenkomplex, so the proper English word is chain complex. https://en.wikipedia.org/wiki/Chain_complex

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`Komplex` means `Subset of a semigroup`, see e.g. Specht, "Die Untergruppen einer Gruppe", Springer 1956. Note that in contrast to a `subgroup`, a Komplex need not be a group (or even a semigroup) itself. I am not aware of an English translation, so you're free to invent one yourself. I should add that until now, I did not come across the term, and as far as I can see, it has been out of fashion for quite some time.

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