A compact Riemann surface
X of genus
g ≥ 2 is called
asymmetric or
pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. The order
d = #(δ) of an anticonformal automorphism δ of such a surface is divisible by 4. In the particular case where
d = 4, δ is a pseudo-symmetry and the surface is called
pseudo-symmetric.
A Riemann surface
X is said to be
p-hyperelliptic if it admits a conformal involution ρ for which the orbit space
X/<ρ> has genus
p. This notion is the particular case of so called
cyclic (
q,
n)-
gonal surface which is defined as the one admitting a conformal automorphism φ of prime order
n such that
X/φ has genus
q. We are interested in possible values of
n and
q for which an asymmetric surface of given genus
g ≥ 2 is (
q,
n)-gonal, and possible values of
p for which the surface is
p-hyperelliptic. Up till now, this problem was solved in the case where the surface is asymmetric and pseudo-symmetric. If an asymmetric Riemann surface
X is not pseudo-symmetric then any anticonformal automorphism of
X has order divisible by 2
sn for
s ≥ 3 and
n = 1 or
n being an odd prime. In this paper we give the necessary and sufficient conditions on the existence of an asymmetric Riemann surface with the full automorphism group being
G =
Z2sn, and we study (
q,
n)-gonal automorphisms and
p-hyperelliptic involutions in
G.
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