Schur-trivial group

Definition
A group is said to be Schur-trivial or a group with trivial Schur multiplier if it satisfies the following equivalent conditions:


 * 1) Its Schur multiplier is the trivial group.
 * 2) The homomorphism from its exterior square to its derived subgroup defined by the commutator map is an isomorphism of groups to the derived subgroup.
 * 3) It is isomorphic to its own Schur covering group with the covering map being the identity map.

Extreme examples

 * The trivial group is Schur-trivial.
 * Cyclic groups are Schur-trivial.

Facts

 * Finite group generated by Schur-trivial subgroups of relatively prime indices is Schur-trivial
 * All Sylow subgroups are Schur-trivial implies Schur-trivial