2-layer

Definition
The 2-layer of a group $$G$$ is defined as follows:

Let $$E_0(G)$$ denote the full inverse image of the layer of $$G/O(G)$$ via the natural projection $$G \to G/O(G)$$. Here $$O(G)$$ denotes the Brauer core of $$G$$, viz the largest normal subgroup of odd order in $$G$$.

The 2-layer of $$G$$ is denoted as $$L(G)$$ or $$E_0^{\infty}(G)$$, and is defined as the subgroup obtained by repeated iteration of the $$E_0$$ operation.