Commutators in finitely generated groups

Posted: February 19, 2010 in Elementary Algebra; Problems & Solutions, Groups and Fields
Tags: , ,

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].

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