Geodesic
Elementary Classes of Groups
Matematičeskie zametki, Tome 72 (2002) no. 1, pp. 84-93

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Let $B$ be a class of groups. The elementary class with base $B$ is defined as the minimal class of groups containing $B$ and closed with respect to taking subgroups, quotient groups, group extensions, and direct limits. Properties of such classes are studied. Some applications to the theory of elementary amenable groups and a relation to the Kurosh–Chernikov classes of generalized solvable groups are considered. 
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     author = {D. V. Osin},
     title = {Elementary {Classes} of {Groups}},
     journal = {Matemati\v{c}eskie zametki},
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D. V. Osin. Elementary Classes of Groups. Matematičeskie zametki, Tome 72 (2002) no. 1, pp. 84-93. http://geodesic.mathdoc.fr/item/MZM_2002_72_1_a7/