Some corollaries of Frobenius' normal -complement theorem

Author:
Yakov Berkovich

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2505-2509

MSC (1991):
Primary 20D20

DOI:
https://doi.org/10.1090/S0002-9939-99-05275-2

Published electronically:
April 28, 1999

MathSciNet review:
1657758

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Abstract | References | Similar Articles | Additional Information

Abstract: For a prime divisor of the order of a finite group , we present the set of -subgroups generating . In particular, we present the set of primary subgroups of generating the last member of the lower central series of . The proof is based on the Frobenius Normal -Complement Theorem and basic properties of minimal nonnilpotent groups. Let be a group and a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non--subgroups (-subgroups) of are not nilpotent. Then (see the lemma), if is generated by all -subgroups of it follows that is a -group.

**[B]**Y. Berkovich, A theorem on nonnilpotent solvable subgroups of finite groups, in `Finite groups', Nauka i Tehnika, Minsk, 1966, pp. 24-39 (Russian).**[Gas]**W. Gaschütz, Über die -Untergruppe endlichen Gruppen, Math. Z. 58 (1953), 160-170. MR**15:285c****[Gol]**Yu.A. Golfand, On groups all of whose subgroups are nilpotent, Dokl. Akad. Nauk SSSR 60 (1948), 1313-1315 (Russian).**[H]**M. Hall, The theory of groups, Macmillan, New York, 1959. MR**21:1996****[Hup]**B. Huppert, Endliche Gruppen, Bd. 1, Springer, Berlin, 1967.**[HB]**B. Huppert and N. Blackburn, Finite Groups II, Springer-Verlag, Berlin, 1982. MR**84i:20001a****[I]**N. Ito, Note on (LM)-groups of finite order, Kodai Math. Seminar Report (1951), 1-6. MR**13:317a****[R]**L. Redei, Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen 4 (1956), 303-324.**[S]**O.Yu. Schmidt, Groups all of whose subgroups are nilpotent, Mat. Sb. 31 (1924), 366-372 (Russian).

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Additional Information

**Yakov Berkovich**

Affiliation:
Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel

Email:
berkov@mathcs2.haifa.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-99-05275-2

Keywords:
Special $p$-group,
minimal nonnilpotent (nonabelian,
noncyclic,
nonsolvable) group,
$p$-nilpotent group,
$p$-closed group,
$\text{S}(p,
q)$-group,
$\text{B}(p,
q)$-group

Received by editor(s):
May 14, 1997

Published electronically:
April 28, 1999

Additional Notes:
The author was supported in part by the Ministry of Absorption of Israel.

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1999
American Mathematical Society