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I'm trying to recall a theorem I saw once about metabelian groups -- it was either of the form "Every metabelian group satisfies..." or of the form "Every group satisfying ... is metabelian".

What was significant/surprising about it was that even though the problem had been open for a while, the proof ultimately just consisted of a few lines of algebraic manipulations (an exception to Scott Aaaronson's number 6).

Unfortunately that's all I remember. Does anyone have any idea what I might be referring to? Thanks!

Edit: It may have been showing that semidirect products of a certain form are metabelian? Not sure. (Now that we have the answer, it wasn't, though this was close.)

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1 Answer 1

up vote 7 down vote accepted

This is a really vague question, and who knows what you saw when, except maybe you, (and you apparently don't). However, one theorem which might fit the bill (it's a nice theorem and it does no harm to publicise it) is that if $G = AB$ for Abelian subgroups $A$ and $B$, not necessarily normal, then the derived group $G^{\prime}$ is Abelian, so $G$ is metabelian. This is a theorem of N.Ito, the proof is a couple of lines, and it had been an open problem for a while.

Noboru Itô, Über das Produkt von zwei abelschen Gruppen, Math. Z. 62(1) (1955), 400-401, MR71426.

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Would it be possible to insert the proof? –  Pierre-Yves Gaillard Aug 6 '11 at 6:17
    
@Pierre-Yves: you can see the first page when clicking on the link in the answer. –  t.b. Aug 6 '11 at 6:28
    
@Theo - Thank you! –  Pierre-Yves Gaillard Aug 6 '11 at 6:51
    
Alternative links: link 1 --- link 2. –  Pierre-Yves Gaillard Aug 6 '11 at 7:20
    
I'm pretty sure this is it. Thank you! –  Harry Altman Aug 6 '11 at 8:09

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