Mathematics – Group Theory
Scientific paper
2010-02-14
M. Chiodo, `Finding non-trivial elements and splittings in groups', J. Algebra. 331, 271-284 (2011)
Mathematics
Group Theory
16 pages, revised version, minor corrections made, additional results on algorithms to construct embeddings and enumerate subg
Scientific paper
10.1016/j.jalgebra.2010.12.013
It is well known that the triviality problem for finitely presented groups is unsolvable; we ask the question of whether there exists a general procedure to produce a non-trivial element from a finite presentation of a non-trivial group. If not, then this would resolve an open problem by J. Wiegold: `Is every finitely generated perfect group the normal closure of one element?' We prove a weakened version of our question: there is no general procedure to pick a non-trivial generator from a finite presentation of a non-trivial group. We also show there is neither a general procedure to decompose a finite presentation of a non-trivial free product into two non-trivial finitely presented factors, nor one to construct an embedding from one finitely presented group into another in which it embeds. We apply our results to show that a construction by Stallings on splitting groups with more than one end can never be made algorithmic, nor can the process of splitting connect sums of non-simply connected closed 4-manifolds.
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