Finite groups with an automorphism of prime order whose fixed points are in the Frattini of a nilpotent subgroup

Anna Luisa Gilotti

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1990)

  • Volume: 1, Issue: 2, page 89-92
  • ISSN: 1120-6330

Abstract

top
In this paper it is proved that a finite group G with an automorphism α of prime order r, such that C G ⁢ α = 1 is contained in a nilpotent subgroup H, with H , r = 1 , is nilpotent provided that either H is odd or, if H is even, then r is not a Fermât prime.

How to cite

top

Gilotti, Anna Luisa. "Finite groups with an automorphism of prime order whose fixed points are in the Frattini of a nilpotent subgroup." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 1.2 (1990): 89-92. <http://eudml.org/doc/244068>.

@article{Gilotti1990,
abstract = {In this paper it is proved that a finite group G with an automorphism \( \alpha \) of prime order r, such that \( C\_\{G\}(\alpha) = 1 \) is contained in a nilpotent subgroup H, with \( (|H|, r) = 1 \), is nilpotent provided that either \( |H| \) is odd or, if \( |H| \) is even, then r is not a Fermât prime.},
author = {Gilotti, Anna Luisa},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Nilpotent subgroup; Automorphism; Simple group; Solvable group; finite group; automorphism of prime order; Frattini subgroup; nilpotent subgroup; finite solvable group},
language = {eng},
month = {5},
number = {2},
pages = {89-92},
publisher = {Accademia Nazionale dei Lincei},
title = {Finite groups with an automorphism of prime order whose fixed points are in the Frattini of a nilpotent subgroup},
url = {http://eudml.org/doc/244068},
volume = {1},
year = {1990},
}

TY - JOUR
AU - Gilotti, Anna Luisa
TI - Finite groups with an automorphism of prime order whose fixed points are in the Frattini of a nilpotent subgroup
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1990/5//
PB - Accademia Nazionale dei Lincei
VL - 1
IS - 2
SP - 89
EP - 92
AB - In this paper it is proved that a finite group G with an automorphism \( \alpha \) of prime order r, such that \( C_{G}(\alpha) = 1 \) is contained in a nilpotent subgroup H, with \( (|H|, r) = 1 \), is nilpotent provided that either \( |H| \) is odd or, if \( |H| \) is even, then r is not a Fermât prime.
LA - eng
KW - Nilpotent subgroup; Automorphism; Simple group; Solvable group; finite group; automorphism of prime order; Frattini subgroup; nilpotent subgroup; finite solvable group
UR - http://eudml.org/doc/244068
ER -

References

top
  1. GILOTTI, A. L., Finite groups with an automorphism of prime order fixing the Frattini subgroup of a Sylow p-subgroup. B.U.M.I., (7) 3-A, 1989. Zbl0679.20020MR1008586
  2. GORENSTEIN, D., Finite groups. Harper & Row, New York1968. Zbl0185.05701MR231903
  3. RICKMAN, B., Groups which admit a fixed point free automorphism oforder p 2 . J. of Algebra, 59, 1979, 77-171. Zbl0408.20017MR541672DOI10.1016/0021-8693(79)90154-6
  4. GORENSTEIN, D., The classification of finite simple groups. I. Bull. of the American Math. Soc., vol. 1, No 1, 1979. Zbl0414.20009MR513750DOI10.1090/S0273-0979-1979-14551-8

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.