## Commutators in finitely generated groups

Posted: February 19, 2010 in Elementary Algebra; Problems & Solutions, Groups and Fields
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Problem. Suppose $G = \langle g_1, \ldots, g_r \rangle$ is a finitely generated group and $N$ is any abelian normal subgroup of $G$. Show that every $g \in [G,N]$ can be written in the form $g = [g_1,n_1]\ldots [g_r,n_r]$ for certain elements $n_1, \ldots, n_r \in N.$

Solution. It follows from the fact that  for all $a_i \in G, n_i \in N$ we have

$[a_1a_2,n_1] = [a_1,a_2n_1a_2^{ - 1}][a_2,n_1]$ and $[a_1, n_1][a_1,n_2] = [a_1,n_1n_2].$