ZJ-subgroup is isoclinism-invariant

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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.