# 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 $G$:

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

## Application to subgroup-defining functions

### 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.