Schur-Baer variety

Definition

A Schur-Baer variety is a subvariety of the variety of groups with the following equivalent conditions:

1. For any group $G$ such that the quotient group of $G$ by its marginal subgroup corresponding to that subvariety is a finite group, it is also true that the verbal subgroup corresponding to that variety is a finite group, and its order divides a power of the order of the quotient by the marginal subgroup.
2. For any finite group $G$, the Baer invariant ${\mathcal {V}}M(G)$ is also a finite group and its order divides a power of the order of $G$ (i.e., all prime factors of its order are also prime factors of the order of $G$).

Equivalence of definitions

Further information: equivalence of definitions of Schur-Baer variety

Facts

• Schur-Baer theorem states that the variety of abelian groups is a Schur-Baer variety.