Outer automorphism groups of metabelian groups

Göbel, Rüdiger; Paras, Agnes T.

doi:10.1016/s0022-4049(99)00022-5

Cited by 15 publications

(9 citation statements)

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“…From Shelah [23, Chapter IV] we adopt the following prediction principle -a model theoretic version of the 'old' and often used Black Box form [24] which was also used and proved in the appendix of [5], see also [11,7,10] for other applications.…”

Section: Appendix: a Model Theoretic Version Of The Black Boxmentioning

confidence: 99%

Constructing Simple Groups for Localizations

Göbel

Shelah

2002

Communications in Algebra

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Abstract. A group homomorphism η : A → H is called a localization of A if every homomorphism ϕ : A → H can be 'extended uniquely' to a homomorphism Φ : H → H in the sense that Φη = ϕ. This categorical concept, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory, see the introduction.For localizations η : A → H of (almost) commutative structures A often H resembles properties of A, e.g. size or satisfying certain systems of equalities and non-equalities.Perhaps the best known example is that localizations of finite abelian groups are finite abelian groups. This is no longer the case if A is a finite (non-abelian) group. Libman showed that A n → SO n−1 (R) for a natural embedding of the alternating group A n is a localization if n is even and n ≥ 10. Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in [12] that any non-abelian finite simple has arbitrarily large localizations. In this paper we want to remove GCH so that the result becomes valid in ordinary set theory.At the same time we want to generalize the statement for a larger class of A's. The new techniques exploit abelian centralizers of free (non-abelian) subgroups of H which constitute a rigid system of cotorsion-free abelian groups. A known strong theorem on the existence of such abelian groups turns out to be very helpful, see [5]. Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg-Mac Lane space K(A, 1) for many groups A. The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13].

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Section: Appendix: a Model Theoretic Version Of The Black Boxmentioning

confidence: 99%

Constructing Simple Groups for Localizations

Göbel

Shelah

2002

Communications in Algebra

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“…Relatives of these groups are constructed in [7] in references given there and for instance in [2,3,5,6]. However they are not good enough for our purpose in this paper.…”

Section: A Results From [8]mentioning

confidence: 99%

“…Here we note that many groups with (a) in the Main Theorem 1.4 are constructed in [6,5,3,2] -in fact for arbitrary groups in place of S ω , but (b), (c) are also crucial. We have an immediate…”

Section: Introductionmentioning

confidence: 99%

Philip Hall's problem on non-Abelian splitters

Göbel

Shelah

2003

Math. Proc. Camb. Phil. Soc.

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Philip Hall raised around 1965 the following question which is stated in the Kourovka Notebook [11, p. 88]: Is there a non-trivial group which is isomorphic with every proper extension of itself by itself ? We will decompose the problem into two parts: We want to find non-commutative splitters, that are groups G = 1 with Ext (G, G) = 1. The class of splitters fortunately is quite large so that extra properties can be added to G. We can consider groups G with the following properties: There is a complete group L with cartesian product L ω ∼ = G, Hom (L ω , S ω ) = 0 (S ω the infinite symmetric group acting on ω) and End (L, L) = Inn L ∪ {0}. We will show that these properties ensure that G is a splitter and hence obviously a Hall-group in the above sense. Then we will apply a recent result from our joint paper [8] which also shows that such groups exist, in fact there is a class of Hall-groups which is not a set.

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“…Notably, Bumagin-Wise proved that every countable group Q can be realised as the outer automorphism group of a finitely generated group G Q [BW05]. Several other authors have achieved results in a similar vein (see, for example, [Koj88], [GP00], [DGG01], [BG03], [FM05], [Min09]). …”

Section: Introductionmentioning

confidence: 87%