Centrally large subgroups permute and their product is centrally large

Statement

Suppose $P$ is a group of prime power order and $A,B$ are Centrally large subgroup (?)s of $P$. Then, $AB=BA$ and $AB$ is also a centrally large subgroup.

Related facts

• All minimal CL-subgroups have the same commutator subgroup
• Centralizer-large subgroups permute and their product is centralizer-large

References

Journal references

• A measuring argument for finite groups by Andrew Chermak and Alberto Delgado, Proceedings of the American Mathematical Society, Volume 107,Number 4, Page 907 - 914(Year 1989): ^{Official copyMore info}
• Centrally large subgroups of finite p-groups by George Isaac Glauberman, Journal of Algebra, ISSN 00218693, Volume 300,Number 2, Page 480 - 508(Year 2006): ^{Official copyMore info}