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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?

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.

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.}

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".