Centrally large subgroup

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This article is about a maximality notion among subgroups, related to abelianness or small class, in a group of prime power order.
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Definition

Let $P$ be a group of prime power order. A subgroup $A$ of $P$ is termed a centrally large subgroup or CL-subgroup if it satisfies the following equivalent conditions:

1. $|A||Z(A)|\geq |B||Z(B)|$ for any subgroup $B$ of $P$.
2. $C_{P}(A)\leq A$ (i.e., $A$ is a self-centralizing subgroup of $P$) and $A$ is a centralizer-large subgroup of $P$.

Equivalence of definitions

Further information: Centrally large iff centralizer-large and self-centralizing

Relation with other properties

Weaker properties

• Self-centralizing subgroup
• Centralizer-large subgroup

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}