Schur-triviality is not quotient-closed
This article gives the statement, and possibly proof, of a group property (i.e., Schur-trivial group) not satisfying a group metaproperty (i.e., quotient-closed group property).
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Example of the quaternion group
This is the smallest example:
- is the quaternion group, which is a Schur-trivial group.
- is the center of quaternion group, which is of order two, and is isomorphic to cyclic group:Z2.
- is isomorphic to the Klein four-group, which is not Schur-trivial.
Perfect group examples
We can construct many examples this way: let be the universal central extension of some perfect group that is not Schur-trivial. Then, is a Schur-trivial group, and is the quotient of by some central subgroup, and is not Schur-trivial.