A Note on Large Characteristic Subgroups

ABSTRACT

It is known that if a group G has an abelian subgroup of finite index n, then it contains an abelian characteristic subgroup A of index at most nⁿ. The aim of this paper is to improve this bound by showing that the characteristic subgroup A can be chosen of index at most n². Examples prove that this bound is the best possible. Our main result is obtained as an application of a general method for the construction of large characteristic subgroups.

KEYWORDS:

• Abelian subgroup
• characteristic subgroup
• twin map

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20E07

Acknowledgements

The authors are members of GNSAGA (INdAM), and work within the AD V-AGTA project.