Finite groups with the pp-embedding property

Stocka, Agnieszka

Rendiconti del Seminario Matematico della Università di Padova, Tome 141 (2019), pp. 107-119.

A subset $X$ of a finite group $G$ is called g\nbdash independent if there is no proper subset $Y$ of $X$ such that $\langle Y,\Phi \left(G\right)\rangle =\langle X,\Phi \left(G\right)\rangle .$ The group $G$ has the embedding property if every g\nbdash independent subset of $G$ can be embedded in a minimal generating set of $G$.If $X$ is a set of prime power order elements, then we say that $G$ has the pp-embedding property. In this note we classify all finite solvable groups with the pp-embedding property. Moreover we prove that this class is equal to the class of finite solvable groups with the embedding property.

Classification : 20, 00
Mots-clés : Finite group, generating set, independent set, soluble group