Fitting classes of $\mathcal{L}_{1}$-groups, I

Beidleman, James C.; Karbe, M. J.; Tomkinson, M. J.

doi:10.1215/ijm/1255988729

Cited by 3 publications

(14 citation statements)

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“…From, say, [9, II.2 and II.6], we see that H l and B { are conjugate in Aut(C/,), so we let w l G Aut(C/ 1 ) be such that (if,)"' 1 = B { . Again assume we have found a suitable w t G Aut(t/ ( ) with (i^,)" 1 Our proof is similar to that of [7,IV. 1].…”

Section: (I) Aut(c/) Has a Unique Non-empty Conjugacy Class Of Sylowmentioning

confidence: 72%

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On minimal Fitting classes of periodic, infinite groups

McCann

1993

J Aust Math Soc A

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Section: (I) Aut(c/) Has a Unique Non-empty Conjugacy Class Of Sylowmentioning

confidence: 72%

“…The class <£ is a subclass of the class of 6 x -groups, for which a theory of injectors has been developed in, for instance, Beidleman, Karbe and Tomkinson [1], Beidleman and Tomkinson [2] and Menegazzo and Newell [10]. Following Menegazzo and Newell, we define a Fitting class of ^-groups to be a subclass X of <S such that:…”

Section: Introductionmentioning

confidence: 99%

On minimal Fitting classes of periodic, infinite groups

McCann

1993

J Aust Math Soc A

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“…(ii) Let G e ft; we show that G XnV is a maximal X n 9)-subgroup of G. Note first that, by Lemma 3.1, X and 9) both contain § n f t and so 3En9)2 §nft. It follows by Theorem 3.1 of [2] that G/G x^v is abelian-by-finite. Let F/G XnV be the Fitting subgroup of G/G XnV so that G/F is finite.…”

Section: Theorem 32 (I)mentioning

confidence: 96%

“…We may therefore assume that 3E does not contain the infinite cyclic group and so 3E consists entirely of Cernikov groups [2,Corollary 2.3]. If ft contains the infinite cyclic group then it contains all finitely generated abelian-by-finite groups and so the example constructed in Theorem 5.1 of [2] shows that there is a ft-group G which does not have I-injectors. This is contrary to X being a normal ft-Fitting class and so we may assume that ft consists entirely of Cernikov groups.…”

Section: Intersections and Products Of Normal Fitting Classesmentioning

confidence: 99%

“…We introduced the notion of a Fitting class of R-groups, or ft-Fitting class, in [2] and, under some rather strong restrictions, obtained an existence and conjugacy theorem for 3E-injectors [2,Theorem 4.4]. A ft-Fitting class X is said to be a normal ft-Fitting class if the radical G x is a maximal 3-subgroup of G, for each Geft, or equivalently, G x is the unique 3£-injector of G, for each Geft.…”

Section: Introductionmentioning

confidence: 99%

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Normal -Fitting classes

Beidleman

Tomkinson²

1992

Proceedings of the Edinburgh Mathematical Society

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The authors together with M. J. Karbe [///. J. Math. 33 (1989) have considered Fitting classes X of S,-groups and, under some rather strong restrictions, obtained an existence and conjugacy theorem for 3£-injectors. Results of Menegazzo and Newell show that these restrictions are, in fact, necessary.The Fitting class X is normal if, for each G e S , , G$ is the unique 3E-injector of G. X is abelian normal if, for each GeS 1 ( GjSG'. For finite soluble groups these two concepts coincide but the class of Cernikov-bynilpotent S,-groups is an example of a nonabelian normal Fitting class of S r groups. In all known examples in which I-injectors exist X is closely associated with some normal Fitting class (the Cernikov-by-nilpotent groups arise from studying the locally nilpotent injectors).Here we investigate normal Fitting classes further, paying particular attention to the distinctions between abelian and nonabelian normal Fitting classes. Products and intersections with (abelian) normal Fitting classes lead to further examples of Fitting classes satisfying the conditions of the existence and conjugacy theorem.

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Fitting classes of CC-groups

Dixon

1988

Proceedings of the Edinburgh Mathematical Society

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The theory of Fitting classes is, by now, a well established part of the theory of finite soluble groups. In contrast, Fitting classes have received rather scant attention in infinite groups, although some recent work of Beidleman and Karbe [2] and Beidleman, Karbe and Tomkinson [3] suggest that one can obtain results in this direction. The paper [2], cited above, in fact generalizes earlier work of Tomkinson [9] to the class of locally soluble FC-groups. The present paper is concerned with the theory of Fitting classes in a class of groups somewhat similar to the class of FC-groups, namely the class of CC-groups, introduced by Polovickiǐ in [6]. A group G is a CC-group if G/CG(xG) is a Černikov group for all x ∈ G where, as in the rest of this paper, we use the standard group theoretic notation of [7]. Recently, Alcázar and Otal [1] have shown how to generalize results of B. H. Neumann [5] to the class of CC-groups. The main purpose of the present note is to illustrate further how one can handle CC-groups, in an analogous manner to FC-groups, by using techniques similar to those used in [1] and [4].

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Fitting classes of $\mathcal{L}_{1}$-groups, I

Cited by 3 publications

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On minimal Fitting classes of periodic, infinite groups

On minimal Fitting classes of periodic, infinite groups

Normal -Fitting classes

Fitting classes of CC-groups

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