A group G is said to be locally graded if every nontrivial, finitely generated subgroup of G has a nontrivial finite image. Every group can occur as a quotient of a locally graded group. It is shown that the largest subgroup and quotient closed interior of the class of locally graded groups is the class of groups in which every simple quotient of every finitely generated subgroup is finite. This article investigates conditions under which a given quotient of a locally graded group is locally graded, and the result is used to get more precise condition for a quotient of a linear group to be locally graded.
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