An integer g >= 2 is said to be a genus of a finite group G if there is a compact Riemann surface of genus g on which G acts as a group of automorphisms. In this paper finite abelian p-groups of arbitrarily large rank, where p is an odd prime, are investigated. For certain classes of abelian p-groups the minimum reduced stable genus sigma(0) of G is calculated and consequently the genus spectrum of G is completely determined for certain "extremal" abelian p-groups. Moreover for the case of Z(p)(r1) circle plus Z(p2)(r2) we will see that the genus spectrum determines the isomorphism class of the group uniquely.


Finite groups having centralizer commutator product property
Ercan, Gülin (2015-12-01)
Let a be an automorphism of a finite group G and assume that G = {[g, alpha] : g is an element of G} . C-G(alpha). We prove that the order of the subgroup [G, alpha] is bounded above by n(log2(n+1)) where n is the index of C-G(alpha) in G.
Torsion Generators Of The Twist Subgroup
Altunöz, Tülin; Pamuk, Mehmetcik; Yildiz, Oguz (2022-1-01)
We show that the twist subgroup of the mapping class group of a closed connected nonorientable surface of genus g >= 13 can be generated by two involutions and an element of order g or g -1 depending on whether 9 is odd or even respectively.
Automorphisms of curve complexes on nonorientable surfaces
Atalan, Ferihe; Korkmaz, Mustafa (2014-01-01)
For a compact connected nonorientable surface N of genus g with n boundary components, we prove that the natural map from the mapping class group of N to the automorphism group of the curve complex of N is an isomorphism provided that g + n >= 5. We also prove that two curve complexes are isomorphic if and only if the underlying surfaces are diffeomorphic.
Automorphisms of complexes of curves on odd genus nonorientable surfaces
Atalan Ozan, Ferihe; Korkmaz, Mustafa; Department of Mathematics (2005)
Let N be a connected nonorientable surface of genus g with n punctures. Suppose that g is odd and g + n > 6. We prove that the automorphism group of the complex of curves of N is isomorphic to the mapping class group M of N.
On abelian group actions with TNI-centralizers
Ercan, Gülin (2019-07-03)
A subgroup H of a group G is said to be a TNI-subgroup if for any Let A be an abelian group acting coprimely on the finite group G by automorphisms in such a way that for all is a solvable TNI-subgroup of G. We prove that G is a solvable group with Fitting length h(G) is at most . In particular whenever is nonnormal. Here, h(G) is the Fitting length of G and is the number of primes dividing A counted with multiplicities.
Citation Formats
E. Y. Talu, “ABELIAN p-GROUPS OF SYMMETRIES OF SURFACES,” TAIWANESE JOURNAL OF MATHEMATICS, pp. 1129–1140, 2011, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/36648.