This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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A group is said to be centralizer-connected if the centralizer of any element is either the whole group or has infinite index in the group., or equivalently, if every conjugacy class with more than one element has infinite size.
Reason for the name
The concept of centralizer-connected is closely related to the concept of connected. In particular, for a connected topological group, every conjugacy class is connected (as it arises as the image of the group under a continuous map). Hence, it must either be a single point or infinite in size, so it must be centralizer-connecteed.
Relation with other properties
These are stronger properties that are satisfied by groups with some additional structure:
In a centralizer-connected group, any finite normal subgroup is central.