# ZJ-subgroup is isoclinism-invariant

## Statement

Suppose $p$ is a prime number. Suppose $P_1$ and $P_2$ are both abelian subgroups of maximum order in $P$ that are isoclinic groups. In particular, there is an isomorphism:

$P_1/Z(P_1) \cong P_2/Z(P_2)$

Combining with the fourth isomorphism theorem, this isomorphism establishes a bijection between the abelian subgroups of $P_1$ containing its center and the abelian subgroups of $P_2$ containing its center.

The claim is that under this bijection, the ZJ-subgroup of $P_1$ maps to the ZJ-subgroup of $P_2$. In particular, the index of $ZJ(P_1)$ in $P_1$ is the same as the index of $ZJ(P_2)$ in $P_2$.