We characterise groups in which every abelian subgroup has finite index in its characteristic closure. In a group with this property every subgroup H has finite index in its characteristic closure and there is an upper bound for this index over all subgroups H of G. For every prime p we construct groups G with this property that are infinite nilpotent p-groups of class 2 and exponent p^2 in which G' = Z(G) is finite and Aut G acts trivially on G/G'. We also characterise abelian groups with the dual property that every subgroup has finite index over its characteristic core.
Finiteness conditions on characteristic closures and cores of subgroups / Cutolo, Giovanni; H., Smith; J., Wiegold. - In: JOURNAL OF GROUP THEORY. - ISSN 1433-5883. - STAMPA. - 12:(2009), pp. 591-610. [10.1515/JGT.2008.092]
Finiteness conditions on characteristic closures and cores of subgroups
CUTOLO, GIOVANNI;
2009
Abstract
We characterise groups in which every abelian subgroup has finite index in its characteristic closure. In a group with this property every subgroup H has finite index in its characteristic closure and there is an upper bound for this index over all subgroups H of G. For every prime p we construct groups G with this property that are infinite nilpotent p-groups of class 2 and exponent p^2 in which G' = Z(G) is finite and Aut G acts trivially on G/G'. We also characterise abelian groups with the dual property that every subgroup has finite index over its characteristic core.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.