Frattini subgroups of 3-manifold groups

Allenby, R. B. J. T.; Boler, J.; Evans, B.; Moser, L. E.; Tang, C. Y.

doi:10.1090/S0002-9947-1979-0517695-8

Frattini subgroups of $3$-manifold groups
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by R. B. J. T. Allenby, J. Boler, B. Evans, L. E. Moser and C. Y. Tang PDF

Trans. Amer. Math. Soc. 247 (1979), 275-300 Request permission

Abstract:

In this paper it is shown that if the Frattini subgroup of the fundamental group of a compact, orientable, irreducible, sufficiently large 3-manifold is nontrivial then the 3-manifold is a Seifert fibered space. We show further that the Frattini subgroup of the group of a Seifert fibered space is trivial or cyclic. As a corollary to our work we prove that every knot group has trivial Frattini subgroup.

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