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I have come to accept that I will never fully understand the formation of compound nouns in German. But I would like some information about one particular example.

Die Kleinsche Vierergruppe is an important structure in the mathematical area of group theory. The name is a compound of "Vier" and "Gruppe", and the extra "er" is something I just have to accept. However, there are some sources that write "Viergruppe". So I wonder if perhaps both forms are used, but one is more common. But then I see this page, which, in a non-mathematical context, uses both forms in the space of four words.

What is going on?

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    I took the liberty to fix the typo in wikipedia. Just in case someone is wonderng what's up in the first link – Christian Mar 11 at 10:05
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For me, "Viergruppe" sounds simply wrong, so I consider it a typo in the sources, you found.

As mentioned in the linked answer compositions of nouns in German may have a "Fugenlaut". For combinations of numbers and other nouns this is indeed the Fugenlaut "-er-", e. g.

  • Dreiergruppe
  • Vierergruppe
  • Fünfergruppe

and so on.

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First of all, note that compounds without -er (Einauge, Zweirad, Dreisatz, Dreiklang, Dreispitz, Vierauge, Fünfkampf) are more frequent than those with -er. I therefore assume compounds with -er have a narrower meaning.

This is borne out by the following observation. There are two ways to form number nouns: By way of zero derivation (feminine, die Eins, Zwei, Drei, etc.) or by adding -er (masculine, der Ein(s)er, Zweier, Dreier, etc.). The latter are more restricted than the former: They are mostly used to refer to grades (ein Fünfer/eine Fünf in Mathe) and money (Ich hab nur noch 'n Fünfer, einen Fünfeuroschein).

The pattern I can discern is that in compounds with -er, the number seems to refer to the internal structure of the object denoted by the head noun.

Zweierzimmer Zimmer mit zwei Betten (Swiss German)
Dreierbeziehung Beziehung zwischen drei Personen
Viererabteil Abteil für vier Personen
Viererpakt Pakt zwischen vier Parteien

Note that these compounds can sometimes be unpacked by adding a noun for the number to count. Then -er vanishes.

Zweibettzimmer, Vierpersonenabteil, Viermächtepakt

The examples you found look like mistakes to me; it should be Vierergruppe in all instances; this is the term Klein used. However, there seem to be some rare (!) cases of missing -er that look genuine. This dissertation from 1908 has Eins-, Zwei-, Drei-, Viergruppe. This could be related to the -er seen as a Fugenlaut and being stigmatized (there once was an attempt to change verfassungsgebend in the German constitution to verfassunggebend).

Having read the other answers to this question, I see that it is highly unclear what kind of -er this is: a nominal suffix, a genitive suffix or a Fugenlaut.

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    Also, in my experience with (older) maths books specifically, there do seem to be some weird nomenclatures around. I can't think of another example right now, but I would not be surprised if some mathematical book used "Viergruppe" consistently (although I would use "Vierergruppe" myself). Finally, what do you make of "Vierkant", "Viereck"... ;) – AnoE Mar 11 at 12:03
  • A Viereck has four corners, so the lack of -er isn't surprising. An interesting case could be Zwei(er)gespann, where the meaning would fit -er but the form without it occurs more frequently. – David Vogt Mar 11 at 12:33
  • Wiktionary notes that the noun Zweier (or Fünfer) referring to a grade is Austrian or perhaps Süddeutsch. Would you agree? – gerrit Mar 11 at 13:15
  • I grew up in Southern Germany and find Fünfer unremarkable. Now we need a Northerner's opinion :) – David Vogt Mar 11 at 13:23
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    @DavidVogt in the North, the grades are referenced as "Eins, Zwei, Drei,..", the Southern variant is extremely uncommon. Regarding the "Zweierzimmer": I once had an annoying experience with an Austrian innkeeper, who referred to his room number 6 as "das Sechserzimmer" on the phone, which I interpreted as "Sechsbettzimmer" (as would be the usual description). There were only 2 beds of course. – Chieron Mar 11 at 15:36
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Zweier, dreier, vierer ... are the genitive case of zwei, drei, vier... A "Vierergruppe" is a group of four. They can be used alone in phrases like "Streit zweier Parteien".

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    "Zweier, dreier, vierer ... are the genitive case of zwei, drei, vier... " doesn't sound right to me. These are just straight nominatives as far as I can tell. "Der 3er BMW" has nothing to do with a Genetiv, it's just the masculine form of "3". Genetive would be "Des Dreiers" / "Des Vierers". – AnoE Mar 11 at 11:59
  • @AnoE. "Zweier" in "Streit zweier Parteien" is not nominative. – fdb Mar 11 at 12:34
  • It's the Genitiv Plural case, isn't it (Wiktionary Flexion:zwei)? Which begs the question: why the plural? – gerrit Mar 11 at 13:11
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    @gerrit. Numbers greater than one are plural by definition. – fdb Mar 11 at 13:14
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    @fdb, exactly, "zweier" is not nominative. But there is a nominative "Zweier" (different word...). You can buy a "Zweier BMW" which just means "the BMW which is labeled with the 2", not "a BMW which is shared between two people". You can pay with a "Zweier" (a coing with the "2 €" on it). I'd say "Vierergruppe" is then made from the Nominative/Substantive "Vierer", not a genetive of "Vier". – AnoE Mar 11 at 14:27

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