A short note on the noncoprime regular module problem

G. Ercan; Ş. Güloğlu

doi:10.37863/umzh.v72i11.6028

G. Ercan Middle East Techn. Univ., Ankara, Turkey
Ş. Güloğlu Doğu¸s Univ., Istanbul, Turkey

DOI: https://doi.org/10.37863/umzh.v72i11.6028

Keywords: nilpotent group, regular orbit, regular module

Abstract

UDC 512.5

Considering a special configuration in which a finite group $A$ acts by automorphisms on а finite group $G$ and the semidirect product $GA$ acts on the vector space $V$ by linear transformations, we discuss the existence of a regular $A$-module in $V_{{A}}.$

References

T. R. Berger, Hall-Higman type theorems , VI, J. Algebra, 51, no. 2, 416 – 424 (1978)., https://doi.org/10.1016/0021-8693(78)90115-1 DOI: https://doi.org/10.1016/0021-8693(78)90115-1

W. Carlip, Regular orbits of nilpotent subgroups of solvable groups , Illinois J. Math., 38, № 2, 199 – 222 (1994).

E. C. Dade, Oral communication to B Huppert , Endliche Gruppen, I, Berlin (1967).

A. Espuelas, The existence of regular orbit , J. Algebra, 127, no. 2, 259 – 268 (1989), https://doi.org/10.1016/0021-8693(89)90251-2 DOI: https://doi.org/10.1016/0021-8693(89)90251-2

A. Espuelas, Regular orbits on symplectic modules , J. Algebra, 138, № 1, 1 – 12 (1991), https://doi.org/10.1016/0021-8693(91)90188-E DOI: https://doi.org/10.1016/0021-8693(91)90188-E

P. Fleischmann, Finite groups with regular orbits on vector spaces , J. Algebra, 103, № 1, 211 – 215 (1986), https://doi.org/10.1016/0021-8693(86)90180-8 DOI: https://doi.org/10.1016/0021-8693(86)90180-8

R. Gow, On the number of characters in a p-block of a $p$-solvable group , J. Algebra, 65, no. 2, 421 – 426 (1980), https://doi.org/10.1016/0021-8693(80)90230-6 DOI: https://doi.org/10.1016/0021-8693(80)90230-6

B. Hargraves, The existence of regular orbits for nilpotent groups , J. Algebra, 72, no. 1, 54 – 100 (1981), https://doi.org/10.1016/0021-8693(81)90312-4 DOI: https://doi.org/10.1016/0021-8693(81)90312-4

B. Huppert, N. Blackburn, Finite Groups , II, Grundlehren Math. Wiss., Springer-Verlag, Berlin, New York, (1982). DOI: https://doi.org/10.1007/978-3-642-67994-0

I. M. Isaacs, Character theory of finite Groups , Dover Publ., Inc., New York, 1994.

A. Turull, Fixed point free action with regular orbits, J. reine und angew. Math., 371, 67 – 91 (1986), https://doi.org/10.1515/crll.1986.371.67 DOI: https://doi.org/10.1515/crll.1986.371.67

A. Turull, Supersolvable automorphism groups of solvable groups, Math. Z., 183, no. 1, 47 – 73 (1983), https://doi.org/10.1007/BF01187215 DOI: https://doi.org/10.1007/BF01187215

Y. Yang, Regular orbits of finite primitive solvable groups, J. Algebra, 323, no. 10, 2735 – 2755 (2010), https://doi.org/10.1016/j.jalgebra.2010.02.031 DOI: https://doi.org/10.1016/j.jalgebra.2010.02.031

Y. Yang, Regular orbits of nilpotent subgroups of solvable linear groups, J. Algebra, 325, 56 – 69 (2011), https://doi.org/10.1016/j.jalgebra.2010.09.024 DOI: https://doi.org/10.1016/j.jalgebra.2010.09.024