# When did the meaning of the term “metabelian” change?

I just realised that the meaning of the term "metabelian", when applied to groups, or Lie algebras, seems to have changed over years. (These days, it means that $[[G,G],[G,G]]$ is trivial, while in the past it was occasionally used to indicate that $[[G,G],G]$ is trivial. The difference here is that between solvability and nilpotence, that is.)

This wiki says "The concept and term metabelian group was introduced by Furtwangler in 1930. The term metabelian was earlier used for groups of nilpotence class 2, but is no longer used in that sense." (I don't understand "earlier" here. Can that sentence be parsed uniquely? Earlier than Furtwangler introduced the term? Earlier than the wiki article was written?)

I know at least one reference from mid 1960s (a PhD thesis from the US) where that old-fashioned usage is present, and I'd like to understand the history better in this instance. Thanks for help.

EDIT: it appears that the situation with usage and its history may be even different for groups and Lie algebras; I only dealt with literature on metabelian Lie algebras, and from answers so far I gather that there may be difference.

• Already Fite (On metabelian groups, Trans. AMS 3 (1902), 331-353) introduced the term metabelian, however not in the modern meaning. Burnside (The most general metabelian group of finite order with two generators, Quart. J. 41 (1910), 223-226) used it for groups with abelian commutator group. – Franz Lemmermeyer Aug 12 '13 at 13:00

I have the impression that even nowadays the term "metabelian" can be confusing if you talk about. Although I would also say, that it means $2$-step solvable now most of the time, it still can mean $2$-step nilpotent sometimes. In particular, "metabelian Lie algebras" can still mean $2$-step nilpotent Lie algebras, see the following (and other) articles:

M. A. Gauger: On the Classification of Metabelian Lie Algebras (1973).
E. M. Luks: What is the typical nilpotent Lie algebra ? (1977).
L. Y. Galitski and D. A. Timashev: On the Classification of Metabelian Lie Algebras (1999).

This is not an answer in the historical sense, but only a comment with too long references (which should perhaps belong to meta, if abelian belongs here).

• I wouldn't say "can still mean" for a paper from 1977 :) I would say that metabelian is confusing; anyway it's not universal in the sense that many people I know outside group theory (but who know what an extension of groups is) are not familiar with this term. – YCor Aug 11 '13 at 22:19
• If you talk to people about meatbelian Lie algebras, they have these articles from 1973, 1977, 1999 etc. in mind, sometimes. In this sense, it still can mean $2$-step nilpotent nowadays. – Dietrich Burde Aug 12 '13 at 7:52

The 1967 book Varieties of Groups by Hanna Neumann has the following footnote (p. 21): The term metabelian will always mean solvable of length two in agreement with current English usage; note however that in much of the Russian literature the term is used in the sense of `nilpotent of class two'.

• Indeed, the two meanings of the word "metabelian" existed side-by-side for many years. I guess that the Russian meaning goes back to Kurosh's book, and the Western meaning - to P. Hall's book (I do not have any of the two books with me now, but it is easy to check). So it is not true that one meaning "evolved" into another one. Simply since Russian mathematicians now publish their papers in Western journals, they use Western terminology. – Mark Sapir Aug 12 '13 at 5:25
• Oh that's most interesting. Thanks. I did not quite realise this aspect: I am mostly interested in metabelian Lie algebras, and all Russians I know always meant 2-step solvable by it, while the PhD thesis I mention in the original question was written in the US. Is it also the case that the history of the term kind of developed independently for groups and for Lie algebras? That'd be amusing. – Vladimir Dotsenko Aug 12 '13 at 7:14
• A quick (but not thorough) check on MathSciNet indicates that almost all of the authors (not just Russians) in the last 10 years or so use the term metabelian Lie algebra for 2-step solvable ones. There might be exceptions. – Primoz Aug 12 '13 at 7:46