Geodesic
Nilpotent Extensions of Abelian p-GROUPS
Canadian journal of mathematics, Tome 38 (1986) no. 5, pp. 1025-1052

Voir la notice de l'article provenant de la source Cambridge University Press

This paper arose out of an attempt to solve the following problem due to Suprunenko [5, Problem 2.77]. For which pairs of abelian groups A, B is every extension of A by B nilpotent? We obtain complete answers when A and B are p-groups and (a) A has finite exponent or (b) B is divisible or (c) A has infinite exponent, is countable and B is non-divisible. The structure of a basic subgroup of A plays a central role in cases (b) and (c).At the outset we must say that the problem is too difficult to solve in complete generality. If G/A ≅ 2?, then the nilpotency of G depends solely on properties of the associated homomorphism θ.B → Aut A. Thus for instance if A is torsion-free and B finite, G is nilpotent if and only if the extension is a central one, and we would need detailed information on finite subgroups of the group Aut A. 
Buckley, Joseph; Wiegold, James. Nilpotent Extensions of Abelian p-GROUPS. Canadian journal of mathematics, Tome 38 (1986) no. 5, pp. 1025-1052. doi: 10.4153/CJM-1986-050-9
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