Schur-Baer variety: Difference between revisions

Latest revision as of 18:16, 31 December 2011

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

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.
For any finite group $G$ , the Baer invariant $V M (G)$ 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 $G$ ).

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

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 ==Definition==
-A '''Schur-Baer variety''' is a subvariety of the [[variety of groups]] with the following property: For any group <math>G</math> such that the [[quotient group]] of <math>G</math> 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.
+A '''Schur-Baer variety''' is a subvariety of the [[variety of groups]] with the following '''equivalent''' conditions:
+# For any group <math>G</math> such that the [[quotient group]] of <math>G</math> 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.
+# For any [[finite group]] <math>G</math>, the [[defining ingredient::Baer invariant]] <math>\mathcal{V}M(G)</math> is also a [[finite group]] and its order divides a power of the order of <math>G</math> (i.e., all prime factors of its order are also prime factors of the order of <math>G</math>).
+===Equivalence of definitions===
+{{further|[[equivalence of definitions of Schur-Baer variety]]}}
 ==Facts==
 * [[Schur-Baer theorem]] states that the variety of [[abelian group]]s is a Schur-Baer variety.