On nilpotent groups of automorphisms of Klein surfaces

Authors:
Emilio Bujalance and Grzegorz Gromadzki

Journal:
Proc. Amer. Math. Soc. **108** (1990), 749-759

MSC:
Primary 20H10; Secondary 14H99, 30F35

DOI:
https://doi.org/10.1090/S0002-9939-1990-0993743-6

MathSciNet review:
993743

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The nilpotent group of automorphisms of a bordered Klein surface of algebraic genus is known to have at most elements. Moreover this bound is attained if and only if is a power of 2. In this paper we prove that if is nonorientable and then the bound in question can be sharpened to which gives a negative answer to a conjecture of May [16]. We also solve another problem of May [16] finding bounds for the -groups of automorphisms of Klein surfaces.

**[1]**E. Bujalance,*Proper periods of normal*NEC*subgroups with even index*, Rev. Math. Hisp.-Amer.**41**(4), 121-127.**[2]**-,*Normal subgroups of*NEC*groups*, Math. Z.**178**(1981), 331-341. MR**635202 (83e:20047)****[3]**E. Bujalance and E. Martinez,*A remark on*NEC*groups of surfaces with boundary*, Bull. London Math. Soc.**21**. (1989), 263-266. MR**986369 (90a:20094)****[4]**E. Bujalance, J. J. Etayo Gordejuela, J. M. Gamboa and G Gromadzki,*A combinatorial approach to automorphism groups of compact bordered Klein surfaces*(in preparation). MR**1075411 (92a:14018)****[5]**J. A. Bujalance,*Normal subgroups of even index in an*NEC*group*, Arch. Math.**49**(1987), 470-478. MR**921112 (89j:20056)****[6]**J. J. Etayo Gordejuela,*Klein surfaces with maximal symmetry and their groups of automorphisms*, Math. Ann.**268**(1984), 533-538. MR**753412 (86g:30058)****[7]**J. J. Etayo Gordejuela and C. Perez-Chirinos,*Bordered and unbordered Klein surfaces with maximal symmetry*, J. Pure Appl. Algebra**42**(1986), 25-35. MR**852315 (88c:14048)****[8]**N. Greenleaf and C. L. May,*Bordered Klein surfaces with maximal symmetry*, Trans. Amer. Math. Soc.**274**(1982), 265-283. MR**670931 (84f:14022)****[9]**A. H. M. Hoare and D. Singerman,*The orientability of subgroups of plane groups*, London Math. Soc. Lecture Note Ser.**71**(1982), 221-227. MR**679163 (85g:20061)****[10]**A. M. Macbeath,*The classification of non-euclidean plane crystallographic groups*, Canad. J. Math.**19**(1967), 1192-1205. MR**0220838 (36:3890)****[11]**C. L. May,*Automorphisms of compact Klein surfaces with boundary*, Pacific J. Math.**59**(1975), 199-210. MR**0399451 (53:3295)****[12]**-,*Large automorphism groups of compact Klein surfaces with boundary*, I, Glasgow Math. J.**111**(1977), 1-10. MR**0425113 (54:13071)****[13]**-,*A bound for the number of automorphisms of a compact Klein surface with boundary*, Proc. Amer. Math. Soc.**63**(1977), 273-280. MR**0435385 (55:8345)****[14]**-,*Maximal symmetry and fully wound coverings*, Proc. Amer. Math. Soc.**79**(1980), 23-31. MR**560577 (81g:14014)****[15]**-,*A family of**-groups*, Canad. J. Math.**37**(1986), 1094-1109.**[16]**-,*Nilpotent automorphism groups of bordered Klein surfaces*, Proc. Amer. Math. Soc.**101**(2) (1987), 287-292. MR**902543 (88j:30093)****[17]**-,*Supersolvable**-groups*, Glasgow Math. J.**30**(1988), 31-40. MR**925557 (89a:20039)****[18]**R. Preston,*Projective structures and fundamental domains on compact Klein surfaces*, Ph.D. thesis, University of Texas, 1975.**[19]**D. Singerman,*as an image of the extended modular group with applications to group actions on surfaces*, Proc. Edinburgh Math. Soc.**30**(1987), 143-151. MR**879440 (88d:20069)****[20]**-,*Automorphisms of compact nonorientable Riemann surfaces*, Glasgow Math. J.**12**(1971), 50-59. MR**0296286 (45:5347)****[21]**M. C. Wilkie,*On non-euclidean crystallographic groups*, Math. Z.**91**(1966), 87-102. MR**0185013 (32:2483)****[22]**R. Zomorrodian,*Nilpotent automorphism groups of Riemann surfaces*, Trans. Amer. Math. Soc.**288**(1985), 241-255. MR**773059 (86d:20059)****[23]**-,*Classification of**-groups of automorphisms of Riemann surfaces and their lower central series*, Glasgow Math. J.**29**(1987), 237-244. MR**901670 (88j:20050)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
20H10,
14H99,
30F35

Retrieve articles in all journals with MSC: 20H10, 14H99, 30F35

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-0993743-6

Keywords:
Klein surfaces,
algebraic genus,
automorphism,
nilpotent group

Article copyright:
© Copyright 1990
American Mathematical Society