CID-operator

Definition
The CID-operator or centralizer-of-involution-domination-operator is an operator that takes as input a subgroup-defining function (which may make sense for all groups or for a more restricted class like finite groups) and outputs a group property as follows. For a subgroup-defining function $$f$$, we say that $$G$$ satisfies the CID of $$f$$ if for any involution $$t$$ in $$G$$:

$$f(C_G(t)) \le f(G)$$

2-layer
The CID-operator applied to the 2-layer is the tautology. In other words, the 2-layer of any centralizer of involution is always contained in the 2-layer of the whole group.

B-subgroup
It is conjectured that the CID-operator applied to the B-subgroup is also the tautology. This is the famous B-conjecture that has not yet been resolved.

Brauer core
The image of the Brauer core subgroup-defining function under the CID-operator is the group property of being a balanced group.