A generalization of a converse to Schur’s theorem
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Niroomand (Arch. Math. 94 (2010) 401–404) proved a converse to a theorem of Schur in the following sense. He proved that if G is a group such that [G, G] is finite and G/Z(G) is finitely generated, then G/Z(G) is finite, of order bounded above by [G, G] k where k is the minimal number of generators required for G/Z(G). Here, we give a completely elementary short proof of a further generalization.