1979
DOI: 10.1090/s0002-9947-1979-0517695-8
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Frattini subgroups of 3-manifold groups

Abstract: 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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Cited by 9 publications
(23 citation statements)
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References 24 publications
(25 reference statements)
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“…Our next result covers some of the cases not covered in [1]. Notice that it is not necessary to assume engulfing here.…”
Section: Hence the Index [H*: H[ Is Finite It Is In Fact The Volume mentioning
confidence: 75%
See 1 more Smart Citation
“…Our next result covers some of the cases not covered in [1]. Notice that it is not necessary to assume engulfing here.…”
Section: Hence the Index [H*: H[ Is Finite It Is In Fact The Volume mentioning
confidence: 75%
“…□ We conclude this section with a slight digression. To place this in context, recall that if G is a group, then the Frattini subgroup of G, usually written $(67), is the intersection of all the maximal subgroups of G. There has been some interest (see [1]) in the Frattini subgroups of knot complements and 3-manifolds.…”
Section: Hence the Index [H*: H[ Is Finite It Is In Fact The Volume mentioning
confidence: 99%
“…Following [1], let C denote the centralizer of N 1 in G 1 , and as in [1], we can deduce various properties about the groups N 1 and C. Namely: (i) since N 1 is a finite group, its centralizer C in G 1 is of finite index in G 1 .…”
Section: Proof Of Theorem 12mentioning
confidence: 99%
“…Its fundamental group π 1 (M) has one of the following three possible presentations, where α i , β i are coprime integers and γ is an integer (see [2] or [14]). …”
Section: A Criterionmentioning
confidence: 99%
“…(1) Free products of conjugacy separable groups are conjugacy separable [17]; (2) The group obtained by adjoining a root to a free group is conjugacy separable [18];…”
Section: Introductionmentioning
confidence: 99%