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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nilpotent automorphism groups of bordered Klein surfaces


Author: Coy L. May
Journal: Proc. Amer. Math. Soc. 101 (1987), 287-292
MSC: Primary 30F35; Secondary 14H99, 20H10
MathSciNet review: 902543
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Abstract: Let $ X$ be a compact bordered Klein surface of algebraic genus $ g \geqslant 2$, and let $ G$ be a group of automorphisms of $ X$. Then the order of $ G$ is at most $ 12(g - 1)$. Here we improve this general bound in an important special case. We show that if $ G$ is nilpotent, then the order of $ G$ is at most $ 8(g - 1)$. This bound is the best possible. We construct infinite families of surfaces that have a nilpotent automorphism group of order $ 8(g - 1)$. The nilpotent groups of maximum possible order must be $ 2$-groups. We prove that if the nilpotent group $ G$ acts on a bordered surface of genus $ g$ such that $ o(G) = 8(g - 1)$, then $ g - 1$ is a power of 2. Further, our examples show that for each nonnegative integer $ t$ there is a bordered surface with genus $ g = {2^t} + 1$ and a group of automorphisms of order $ 8(g - 1)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0902543-4
PII: S 0002-9939(1987)0902543-4
Keywords: Bordered Klein surface, genus, automorphism, nilpotent group, $ {M^ * }$-group, full covering
Article copyright: © Copyright 1987 American Mathematical Society