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Fundamental group of a 3-manifold. (English. Russian original) Zbl 0721.57001
Algebra Logic 29, No. 3, 228-231 (1990); translation from Algebra Logika 29, No. 3, 345-349 (1990).
Theorem: The group $$G(p)=<x,y|$$ $$x^ 3=y^ 3$$, $$x^{3p}=(zx^{- 1})^ 2>$$ has order 72p for every $$p\geq 1$$. These groups are the fundamental groups of certain 3-manifolds what makes them interesting. For odd p or $$p\leq 11$$ the above result has been obtained by Lonergan- Hosack. The main tool for the proof is the Reidemeister-Schreier method for calculating the commutator subgroup. The factor group is cyclic of order 9p and the commutator subgroup is isomorphic to the group $$<v_ 0,v_ 1,v_ 2|$$ $$v_ 0v_ 1=v_ 2$$, $$v_ 1v_ 2=v_ 0$$, $$v_ 2v_ 0=v_ 1$$, $$[v^ 2_ i,v_ j]=1$$ $$0\leq i,j\leq 2>$$ of order 8 (the multiplicative group generated by the quaternions of order 4).
##### MSC:
 57M05 Fundamental group, presentations, free differential calculus 20F05 Generators, relations, and presentations of groups
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##### References:
 [1] F. D. Londergan and J. M. Hosack, ”Examples of 3-manifolds with finite fundamental groups. III,” in: Abstracts of Papers Presented to American Math. Society,9, No. 5, 426 (1988). [2] M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of group Theory [in Russian], Nauka, Moscow (1982). · Zbl 0508.20001 [3] W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Interscience, New York (1966).
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