SUBGROUP LATTICE ALGORITHMS RELATED TO EXTENDING AND LIFTING ABELIAN GROUPS

We develop algorithms for determining properties of finite abelian groups related to the notions of extending and lifting groups. Thus, we give efficient methods, on one hand to check the properties of being direct summand, essential, superfluous, coessential, complement (closed), supplement (coclosed) subgroup, and on the other hand to determine all subgroups with the mentioned properties of a given finite abelian group.

Keywords:

Subgroup lattice, essential subgroup, superfluous subgroup, complement, supplement extending abelian group, lifting abelian group,

PDF

___

G. C˘alug˘areanu, S. Breaz, C. Modoi, C. Pelea and D. V˘alcan, Exercises in Abelian group theory, Kluwer Texts in the Mathematical Sciences, 25, Dor- drecht, Kluwer, 2003.
J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules. Supplements and projectivity in module theory. Frontiers in Mathematics, Birkh¨auser, Basel, S. Crivei, G. Olteanu and S¸. S¸uteu Sz¨oll˝osi, ELISA. A collec- tion of GAP algorithms related to extending and lifting abelian groups. (http://www.gap-system.org/Packages/undep.html) (http://math.ubbcluj.ro/~crivei/GAP_project). N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Pitman Research Notes in Mathematics Series, 313, Longman ScientiŞc and Technical, 1994.
A. Harmancı, D. Keskin and P.F. Smith, On ⊕-supplemented modules, Acta Math. Hungar., 83 (1999), 161–169.
D. Keskin and W. Xue, Generalizations of lifting modules, Acta Math. Hun- gar., 91 (2001), 253–261.
E. Mermut, Homological approach to complements and supplements, Ph.D. thesis, Dokuz Eyl¨ul University, Izmir, 2004.
S.H. Mohamed and B.J. M¨uller, Continuous and discrete modules, London Math. Soc. Lecture Notes Series, 147, Cambridge Univ. Press, Cambridge, P.F. Smith and A. Tercan, Generalizations of CS modules, Comm. Algebra, (1993), 1809–1847.
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7 ; 2006, (http://www.gap-system.org).
Septimiu Crivei * and S¸tefan S¸uteu Sz¨oll˝osi ** Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, Str. M. Kog˘alniceanu 1, 400084 Cluj-Napoca, Romania
E-mails: * crivei@math.ubbcluj.ro, ** szollosi@gmail.com