# ZJ-subgroup is isoclinism-invariant

From Groupprops

## Statement

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

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

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