Finite groups with a Frobenius subgroup

A. V. Romanovskii

Sbornik. Mathematics, Tome 36 (1980) no. 4, pp. 577-601

Résumé

Suppose $M$ denotes a $CC$-subgroup of order $m$ of a group $G$ which is different from its normalizer in $G$. A criterion for the simplicity of a group is obtained which includes the theorems of Feit and Ito on Zassenhaus groups of even degree and which is used to prove the following Theorem. If $|G:N(M)|=m+1$ and the order of the centralizer of each nonidentity element of $N(M)$ in $G$ is odd, then $G\simeq PSL(2,m)$. It is proved that if $M$ has a complement $B$ in $G$ and if $|M|-1$ does not divide $|B|$, then $N(M)$ has a nilpotent invariant complement in $G$, and if $M$ is complemented by a Frobenius subgroup in the simple group $G$, then $G\simeq PSL(2,2^n)$, $n>1$. Related to the results of Brauer, Leonard, and Sibley on finite linear groups is the following Theorem. {\it If the degree of each irreducible constituent of some faithful complex character $\varphi$ of $G$ is less than $(m-1)/2$, then either $M\lhd G$ or $G\simeq Sz(2^{2n+1})$, $n\geqslant1$.} Other results connected with the above theorems are also obtained. Bibliography: 24 titles.

Bibliographie

[1] A. I. Kostrikin, Vvedenie v algebru, izd-vo “Nauka”, Moskva, 1977 | MR | Zbl

[2] A. I. Kostrikin, “Konechnye gruppy”, Algebra, 1964. Itogi nauki, VINITI AN SSSR, Moskva, 1966, 7–46 | MR

[3] S. A. Chunikhin, Podgruppy konechnykh grupp, izd-vo “Nauka i tekhnika”, Minsk, 1964 | MR

[4] V. D. Mazurov, “Konechnye gruppy”, Itogi nauki. Algebra. Topologiya. Geometriya, 14, VINITI AN SSSR, Moskva, 1976, 5–56

[5] V. A. Belonogov, A. N. Fomin, Matrichnye predstavleniya v teorii konechnykh grupp, izd-vo “Nauka”, Moskva, 1976 | MR

[6] A. V. Romanovskii, “O konechnykh gruppakh s podgruppoi Frobeniusa”, Matem. zametki, 20:2 (1976), 177–186 | MR | Zbl

[7] A. V. Romanovskii, “Ob invariantnom dopolnenii k podgruppe Frobeniusa”, DAN BSSR, 21:4 (1977), 293–295 | MR | Zbl

[8] A. V. Romanovskii, “K teoremam Feita i Ito o gruppakh Tsassenkhauza”, DAN BSSR, 22:4 (1978), 293–295 | MR | Zbl

[9] M. Kholl, Teoriya grupp, IL, Moskva, 1962

[10] R. Brauer, H. S. Leonard, “On finite groups with an abelian Sylow group”, Canad. J. Math., 14:3 (1962), 436–450 | MR | Zbl

[11] W. Feit, Characters of finite groups, New York, 1967 | MR

[12] W. Feit, “On a class of doubly transitive permutation groups, III”, J. Math., 4:2 (1960), 170–186 | MR | Zbl

[13] D. Gorenstein, Finite groups, New York, 1968 | Zbl

[14] M. Herzog, “On finite groups which contain a Frobenius subgroup”, J. Algebra, 6:2 (1967), 192–221 | DOI | MR | Zbl

[15] D. F. Holt, “Doubly transitive groups with a solvable one point stabilizer”, J. Algebra, 44:1 (1977), 29–92 | DOI | MR | Zbl

[16] B. Huppert, Endliche Gruppen. I, Springer-Verlag, Berlin–Heidelberg–New York, 1967 | MR | Zbl

[17] N. Ito, “On a class of doubly transitive permutation groups, III”, J. Math., 6:2 (1962), 341–352 | MR | Zbl

[18] H. S. Leonard, “On finite groups which contain a Frobenius factor group, III”, J. Math., 9:l (1965), 47–58 | MR

[19] H. S. Leonard, “Finite linear groups having an abelian Sylow subgroup. II”, J. Algebra, 26:2 (1973), 368–382 | DOI | MR | Zbl

[20] D. A. Sibley, “Finite linear groups with a strongly self-centralizing Sylow subgroup. II”, J. Algebra, 36:2 (1975), 319–332 | DOI | MR | Zbl

[21] D. A. Sibley, “Coherence in finite groups containing a Frobenius section, III”, J. Math., 20:3 (1976), 434–442 | MR | Zbl

[22] M. Suzuki, “On a class of doubly transitive groups”, Ann. Math., 75:1 (1962), 105–145 | DOI | MR | Zbl

[23] M. Suzuki, “Two characteristic properties of $ZT$-groups”, Osaka Math. J., 15:1 (1963), 143–150 | MR | Zbl

[24] H. Zassenhaus, “Kennzeichnung endlicher linear Gruppen als Permutationsgruppen”, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 17–40 | DOI