Abstract
A non-nilpotent finite group whose proper subgroups are all nilpotent (or a finite group without non- nilpotent proper subgroups) is well-known (called Schmidt group). O.Yu. Schmidt (1924) studied such groups and proved that such groups are solvable. More recently Zarrin generalized Schmidt's Theorem and proved that every finite group with less than 22 non-nilpotent subgroups is solvable. In this paper, we show that every locally graded group with less than 22 non-nilpotent subgroups is solvable.
Highlights
Introduction and ResultsLet G be a group
A group G is said to be locally graded if every non-trivial finitely generated subgroup of G has a non-trivial finite homomorphic image
It is well known that if G is a group and H is a normal subgroup of G such that H is solvable and G/H is solvable G is solvable Proof of Theorem 1.1
Summary
Introduction and ResultsLet G be a group. A non-nilpotent finite group whose proper subgroups are all nilpotent (or a finite group without non-nilpotent proper subgroups) is well-known (calledSchmidt group). Keywords Schmidt Group, Locally Graded Group, Solvable Group A group G is said to be locally graded if every non-trivial finitely generated subgroup of G has a non-trivial finite homomorphic image. A non-abelian finite group whose proper subgroups are all abelian is well-known (called Miller-Moreno group).
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