Centrally closed subgroup

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]

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Definition

Symbol-free definition

A subgroup of a group is termed centrally closed or a CC-subgroup if the centralizer of any non-identity element of the subgroup lies inside the subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed centrally closed or a CC-subgroup if for any $h$ in $H$, the group $C_G(h)$ lies inside $H$.

Relation with other properties

Stronger properties

• Malnormal subgroup
• Frobenius kernel

Weaker properties

• Self-centralizing subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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If $H$ is a CC-subgroup of $K$ and $K$ is a CC-subgroup of $G$, then $H$ is a CC-subgroup of $G$. The proof of this follows directly from the definitions.

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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The trivial subgroup is clearly CC, and so is the whole group. Thus, the property of being a CC-subgroup is also a t.i. subgroup property

Note that proper nontrivial CC-subgroups can occur only in a centerless group.

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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Any subgroup that is CC in the whole group, is also CC in any intermediate subgroup. This is because the centralizer in any intermediate subgroup is contained inside the centralizer in the whole group.