# Trying to recall a theorem about metabelian groups with a surprisingly simple proof

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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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.