It is stated in $\S$1 of Romankov's:
Generators of Automorphisms of Free Metabelian pro-$p$ groups
that if $M$ is a metabelian group, $u\in M'$, then for any $g,f$, we have $[[u,g],f] = [[u,f],g]$.
Is this true? I can't seem to verify this...
In a metabelian group $M$, is it true that $[[u,g],f] = [[u,f],g]$ for $u\in M'$, $g,h\in M$?
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group-theory
1 Answers
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Yes, this is well-known, and a proof can be found in many references, e.g., in Lemma 3.1. at page $7$ here. It states that for a metabelian group $G$ we have $$ [c,b,a]=[c,a,b] $$ for all $a\in G'=[G,G]$, with the notation $[a,b,c]=[[a,b],c]$. For the proof one first shows the Hall-Witt identity $$ [c, b, a] = [b, a, c]^{−1} [c, a, b] $$ which holds for all groups, and then derives the result as a corollary.