Non-nilpotent Subgroups in Locally Graded Groups

A non-nilpotent finite group whose proper subgroups are all nilpotent (or a finite group without nonnilpotent 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.


Introduction and Results
Let 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 (called Schmidt group). O. Yu. Schmidt (1924) studied such groups and proved that such groups are solvable [4]. Subsequently, Newman and Wiegold in [1], discuss infinite non-nilpotent groups whose proper subgroups are all nilpotent. Such groups need not be solvable in general. For example, the T arski M onsters, which are infinite simple groups with all proper subgroups of a fixed prime order. Following [7] we say that a group G is a S n -group if G has exactly n nonnilpotent subgroups. Also we say that G has a finite number of non-nilpotent subgroups, written G ∈ S, if G ∈ S n for some n ∈ N. Indeed S = ∪ i≥1 S i . The S n -groups G with n = 1, namely non-nilpotent groups all of whose proper subgroups are nilpotent (i.e. minimal non-nilpotent groups). Therefore Schmidt's Theorem says that every finite S 1 -group is solvable. More recently Zarrin in [7] generalized Schmidt's Theorem and proved that every finite S n -group with n < 22 is solvable. But this result is not true for infinite S n -groups (for instance the T arski M onsters groups are infinite S 1 -groups).
Here we improve this result as follows: 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. This is a rather large class of groups, containing for instance all residually finite groups and all locally(soluble by finite) groups. It is often considered in order to avoid T arski monsters groups.
A non-abelian finite group whose proper subgroups are all abelian is well-known (called Miller-Moreno group). In 1903, Miller and Moreno pioneered the study of such groups and proved that such groups are solvable [3]. Now, using the above result and as every non-nilpotent subgroup is a non-abelian subgroup, we generalize their result as follows.
Corollary 1.2. Every locally graded (or, locally solvable) group with less than 22 non-abelian subgroups is solvable.

Proof
For the proof of main Theorem, we using similar methods as [8]. 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 then G is solvable Proof of Theorem 1.1.
Suppose that G is a S n -group with n ≤ 21.
The group G acts on the set X by conjugation. Now the subgroup is finite and also, by Lemma 2.3 of [7], G AN (G) is a S r -group with r < 22. Therefore, according to the main result of [7], to complete the proof it is enough to show that AN (G) is solvable. To see this, suppose that K is a non-nilpotent of AN (G). It follows that K is a non-nilpotent of G and so, by definition of AN (G), we obtain that K ▹ AN (G). Therefore every non-nilpotent subgroup of AN (G) is normal (and hence is subnormal). If G is a locally graded group, then AN (G) is also locally graded group and so, by Theorem 2 of [5], it follows that AN (G) is solvable. If G is a locally solvable group, then AN (G) is also locally solvable group and so, by the main result of [2], AN (G) either is locally nilpotent or has finite commutator subgroup. Now if AN (G) is locally nilpotent, then it follows from Lemma 2.2 of [5], that AN (G) is a solvable group. Finally, if AN (G) ′ the commutator subgroup of AN (G) is finite then the main result of [7] follows that AN (G) ′ is a solvable and so AN (G) is solvable and this completes the proof.
Remark 2.1. In view of the proof of the Theorem, we can see that every locally graded (or, locally solvable) S-group is solvable-by-finite. More precisely, if G is an arbitrary group with a finite number n of non-nilpotent subgroups, then the factor group G AN (G) is finite and | G AN (G) |≤ n!. This result suggests that the behaviour of non-nilpotent subgroups has a strong influence on the structure of the group.
Smith in [6] showed that every torsion-free locally solvable-by-finite group in which every non-nilpotent subgroup is subnormal is nilpotent. Now by a similar argument as in the proof of the Theorem, mentioned for AN (G), we can give the following Corollary.

Corollary 2.2. Every torsion-free locally solvable-by-finite group with a finite number of non-nilpotent subgroups is nilpotent-by-finite.
Proof. Assume that G is a torsion-free locally solvable-byfinite group with a finite number of non-nilpotent subgroups. Then, according to Remark 2.1, the factor group G AN (G) is finite and the subgroup AN (G) is solvable since every locally solvable-by-finite group is locally graded group. On the other hand, AN (G) is torsion-free locally solvable-by-finite group and also, as we mentioned in the proof of main Theorem, every non-nilpotent subgroup of AN (G) is normal. There-fore, by [6], one can follow that AN (G) is nilpotent and this completes the proof.