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In this paper we investigate the situation where a group G has an abelian subgroup H with connected transversals. We show that if H is finite then G is solvable. We also investigate some special cases where the structure of H is very close to the structure of a cyclic group. Finally we apply our results to loop theory and we show that if the inner mapping group of a finite loop Q is abelian then Q is centrally nilpotent.

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In this article we show that finite loops with nilpotent inner mapping groups are centrally nilpotent.2000 Mathematics subject classification: primary 20D10, 20N05.