Centrally large subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
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)| \ge |B||Z(B)| for any subgroup B of P.
  2. C_P(A) \le 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

References

Journal references