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On soluble groups of automorphisms on nonorientable Klein surfaces. (English) Zbl 0822.20033

Soluble groups of automorphisms of a compact Riemann surface of genus \(q\) are well-known to have order bounded by \(48 (q - 1)\). For compact non- orientable Klein surfaces without boundary and algebraic genus \(q\), the maximal order of a soluble group of automorphisms is shown here to be \(24 (q-1)\) and this bound is attained for infinitely many values of \(q\). The group theoretic formulation of this problem also shows that those groups for which this maximum is attained occur as maximal symmetry groups for non-orientable Klein surfaces with \(q - 1\) boundary components and algebraic genus \(2q - 1\) [cf. C. L. May, Glasg. Math. J. 18, 1-10 (1977; Zbl 0363.14008)]. The topological classification of non-orientable bordered Klein surfaces with maximal symmetry can be given a group theoretic interpretation, and the author determines this classification in the case where the maximal group is soluble with derived length 4.

MSC:

20F05 Generators, relations, and presentations of groups
30F10 Compact Riemann surfaces and uniformization
30F50 Klein surfaces
20F16 Solvable groups, supersolvable groups
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks

Citations:

Zbl 0363.14008
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