Exponent of a finite group admitting a coprime automorphism of prime order
-
Sara Rodrigues
Abstract
Let πΊ be a finite group admitting an automorphism π of prime order π such that
Funding statement: This work was supported by the Conselho Nacional de Desenvolvimento CientΓfico e TecnolΓ³gico (CNPq) and Fundação de Apoio Γ Pesquisa do Distrito Federal (FAPDF), Brazil.
Communicated by: Evgenii I. Khukhro
References
[1] B. Bruno and F. Napolitani, A note on nilpotent-by-Chernikov groups, Glasg. Math. J. 46 (2004), no. 2, 211β215. 10.1017/S0017089504001703Search in Google Scholar
[2] F. de Giovanni and M. Trombetti, Large characteristic subgroups with restricted conjugacy classes, Results Math. 74 (2019), no. 4, Paper No. 166. 10.1007/s00025-019-1089-5Search in Google Scholar
[3] D. Gorenstein, Finite Groups, Harper & Row, New York, 1968. Search in Google Scholar
[4] G. Higman, Groups and rings having automorphisms without non-trivial fixed elements, J. Lond. Math. Soc. 32 (1957), 321β334. 10.1112/jlms/s1-32.3.321Search in Google Scholar
[5] E.βI. Khukhro, Groups and Lie rings admitting almost regular automorphisms of prime order (in Russian), Mat. Sb. 181 (1990), 1197β1219; English translation in Math. USSR Sb. 71 (1992), 51-63. 10.1070/SM1992v071n01ABEH001390Search in Google Scholar
[6] E.βI. Khukhro, Groups with an automorphism of prime order that is almost regular in the sense of rank, J. Lond. Math. Soc. (2) 77 (2008), no. 1, 130β148. 10.1112/jlms/jdm097Search in Google Scholar
[7] E.βI. Khukhro and N.βY. Makarenko, Large characteristic subgroups satisfying multilinear commutator identities, J. Lond. Math. Soc. (2) 75 (2007), no. 3, 635β646. 10.1112/jlms/jdm047Search in Google Scholar
[8] E.βI. Khukhro and V.βD. Mazurov, Finite groups with an automorphism of prime order whose centralizer has small rank, J. Algebra 301 (2006), no. 2, 474β492. 10.1016/j.jalgebra.2006.02.039Search in Google Scholar
[9] E.βI. Khukhro and P. Shumyatsky, Nonsoluble and non-π-soluble length of finite groups, Israel J. Math. 207 (2015), no. 2, 507β525. 10.1007/s11856-015-1180-xSearch in Google Scholar
[10] E.βI. Khukhro and P. Shumyatsky, On the length of finite groups and of fixed points, Proc. Amer. Math. Soc. 143 (2015), no. 9, 3781β3790. 10.1090/S0002-9939-2015-12573-7Search in Google Scholar
[11] A. Mann, The exponents of central factor and commutator groups, J. Group Theory 10 (2007), no. 4, 435β436. 10.1515/JGT.2007.035Search in Google Scholar
[12] D.βJ.βS. Robinson, A Course in the Theory of Groups, 2nd ed., Grad. Texts in Math. 80, Springer, New York, 1996. 10.1007/978-1-4419-8594-1Search in Google Scholar
[13] S. Rodrigues and P. Shumyatsky, Exponent of a finite group of odd order with an involutory automorphism, Arch. Math. (Basel) 113 (2019), no. 2, 113β118. 10.1007/s00013-019-01318-5Search in Google Scholar
[14] S. Rodrigues and P. Shumyatsky, Exponent of a finite group admitting a coprime automorphism, J. Pure Appl. Algebra 224 (2020), no. 9, Article ID 106370. 10.1016/j.jpaa.2020.106370Search in Google Scholar
[15] J. Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Natl. Acad. Sci. USA 45 (1959), 578β581. 10.1073/pnas.45.4.578Search in Google Scholar
[16] J.βG. Thompson, Automorphisms of solvable groups, J. Algebra 1 (1964), 259β267. 10.1016/0021-8693(64)90022-5Search in Google Scholar
[17] A. Turull, Character theory and length problems, Finite and Locally Finite Groups (Istanbul 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 471, Kluwer Academic, Dordrecht (1995), 377β400. 10.1007/978-94-011-0329-9_14Search in Google Scholar
Β© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The groups (2, π | π, π | 1, π): Finiteness and homotopy
- Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem
- Metabelian groups: Full-rank presentations, randomness and Diophantine problems
- Nilpotence relations in products of groups
- The subnormal structure of classical-like groups over commutative rings
- A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups
- On the stabilisers of points in groups with micro-supported actions
- Groups that have a partition by commuting subsets
- Commuting graph of π΄-orbits
- On the holomorph of finite semisimple groups
- TI subgroups and depth 3 subgroups in simple Suzuki groups
- On the minimal degree of a transitive permutation group with stabilizer a 2-group
- Exponent of a finite group admitting a coprime automorphism of prime order
Articles in the same Issue
- Frontmatter
- The groups (2, π | π, π | 1, π): Finiteness and homotopy
- Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem
- Metabelian groups: Full-rank presentations, randomness and Diophantine problems
- Nilpotence relations in products of groups
- The subnormal structure of classical-like groups over commutative rings
- A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups
- On the stabilisers of points in groups with micro-supported actions
- Groups that have a partition by commuting subsets
- Commuting graph of π΄-orbits
- On the holomorph of finite semisimple groups
- TI subgroups and depth 3 subgroups in simple Suzuki groups
- On the minimal degree of a transitive permutation group with stabilizer a 2-group
- Exponent of a finite group admitting a coprime automorphism of prime order
