Abstract
Subgrouppermutability in a finite group is studied, in order to distinguish between irreducible and reducible groups and to give a bound for the diameter. A simple nonabelian finite group is irreducible and its diameter is less or equal16 while a reducible group is forced to be soluble and the bound of the diameter of such a group is 4.
Article PDF
Similar content being viewed by others
References
M.Aschbacher,Simple connectivity of p-group complexes, to appear in Israel J. Math.
M. Bianchi -D. Chillag -A. Gillio -M. Herzog -C. M. Scoppola,Application of a graph related to conjugacy classes in finite groups, Arch. Math.,58 (1992), pp. 126–132.
M. Bianchi -A. Gillio -L. Verardi,Groups with small subgroup-permutability diameter, Boll. Un. Mat. Ital. (7),6-B (1992), pp. 229–239.
M. Bianchi -A. Gillio -L. Verardi,p-groups in which subgroup-permutability is transitive, Boll. Un. Mat. Ital. (7),9-B (1995), pp. 61–76.
R. W. Carter,Conjugacy classes in the Weyl group, Comp. Math.,25, Fasc. I (1972), pp. 1–59.
J. H. Conway -R. T. Curtis -S. P. Norton -R. A. Parker -R. A. Wilson,Atlas of Finite Groups, Clarendon Press, Oxford (1985).
L. Di Martino -M. C. Tamburini, 2-generationof finite simple groups and some related topics, inGenerators and Relations in Groups and Geometries, Kluwer Academic Publ., Netherlands (1991), pp. 195–233.
R. Gow,Product of two involutions in classical groups of characteristic 2, J. Algebra,71 (1981), pp. 583–591.
H. Heineken -H. Liebeck,The occurrence of finite groups in the automorphism group of nilpotent groups of class 2, Arch. Math.,25 (1974), pp. 8–16.
A. R. Hoffer,On unitary collineation groups, J. Algebra,22 (1972), pp. 211–218.
B. Huppert,Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York (1967).
K. Iwasawa,Uber die endlichen Gruppen und die Verbande ihrer Untergruppen, J. Univ. Tokyo,4-3 (1941), pp. 171–199.
P. B. Kleidman,The maximal subgroups of the Steinberg triality groups 3 D 4 (q) and their automorphism groups, J. Algebra,115 (1988), pp. 182–199.
M. W. Liebeck -C. E. Praeger -J. Saxl,A classification of the maximal subgroups of finite alternating and symmetric groups, J. Algebra,111 (1987), pp. 365–383.
G. Malle,The maximal subgroups of 2 F 4(q 2), J. Algebra,139 (1991), pp. 52–69.
K. Mizuno,The conjugacy classes of Chevalley groups of type E 6, J. Fac. Sci. Univ. Tokyo,24 (1977), pp. 525–563.
E. Stensholt,Certain embeddings among finite groups of Lie type, J. Algebra,53 (1978), pp. 136–187.
M. Suzuki,On a class of doubly transitive groups, Ann. Math.,75, n. 1 (1962), pp. 105–145.
C. Tibiletti Marchionna,Sulla distanza di un gruppo, R.I.S.L. A,112 (1978), pp. 181–191.
C. Tibiletti Marchionna,Sui gruppi a distanza d⩽2, Boll. Un. Mat. Ital. (5),17-B (1980), pp. 14–32.
H. N. Ward,On Ree's series of simple groups, Trans. Amer. Math. Soc.,121 (1966), pp. 62–89.
T. S. Weigel,Generation of exceptional groups of Lie-type, Geom. Dedicata,41, n. 1 (1992), pp. 63–87.
J. S. Williams,Prime graph components of a finite group, J. Algebra,69 (1981), pp. 487–513.
W. Wong,A characterization of the finite projective symplectic group PSp4 (q), Trans. Amer. Math. Soc.,139 (1969), pp. 1–35.
W. Wong,A characterization of the finite simple groups PSp 2n (q), Trans. Amer. Math. Soc.,139 (1969), pp. 531–555.
Author information
Authors and Affiliations
Additional information
Research partially supported by G.N.S.A.G.A. of C.N.R. and M.U.R.S.T. of Italy.
Rights and permissions
About this article
Cite this article
Bianchi, M., Gillio Berta Mauri, A. & Verardi, L. Finite groups and subgroup-permutability. Annali di Matematica pura ed applicata 169, 251–268 (1995). https://doi.org/10.1007/BF01759356
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01759356