Abstract A group G is said to be n -central if the factor group G/Z ( G ) is of exponent n . We improve a result of Gupta and Rhemtulla by showing that every 4-central group is 16-abelian and every 6-central group is 36-abelian. There are examples of finite groups which show that these bounds are best possible. Consequently, we can completely describe the structure of exponent semigroups of free non-cyclic n -central groups for n = 2, 3, 4, 6. We obtain a characterization of metabelian p -central groups and a classification of finitely generated 2-central groups. We compute the nilpotency class of the free metabelian 4-central group of arbitrary finite rank.
Journal of Group Theory – de Gruyter
Published: Jul 1, 2006
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