Baumann subgroup corresponding to join of abelian subgroups of maximum order

Definition
Let $$p$$ be a prime number and $$P$$ be a finite p-group. The Baumann subgroup of $$P$$, denoted $$B(P)$$, is defined as the defining ingredient::centralizer in $$P$$ of the defining ingredient::first omega subgroup of the defining ingredient::ZJ-subgroup of $$P$$. In symbols:

$$\! B(P) := C_P(\Omega_1(Z(J(P))))$$

Here, the ZJ-subgroup refers to the center of the join of abelian subgroups of maximum order.

Relation with other subgroup-defining functions
See also Baumann subgroup corresponding to join of elementary abelian subgroups of maximum order.