Centrally closed subgroup
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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 of a group is termed centrally closed or a CC-subgroup if for any in , the group lies inside .
Relation with other properties
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 is a CC-subgroup of and is a CC-subgroup of , then is a CC-subgroup of . The proof of this follows directly from the definitions.
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.