Abstract
LetG be an arbitrary group with a subgroupA. The subdegrees of (A, G) are the indices [A:A ∪A 9] (wheregεG). Equivalent definitions of that concept are given in [IP] and [K]. IfA is not normal inG and all the subdegrees of (A, G) are finite, we attach to (A, G) the common divisor graph Γ: its vertices are the non-unit subdegrees of (A, G), and two different subdegrees are joined by an edge iff they arenot coprime. It is proved in [IP] that Γ has at most two connected components. Assume that Γ is disconnected. LetD denote the subdegree set of (A, G) and letD 1 be the set of all the subdegrees in the component of Γ containing min(D−{1}). We proved [K, Theorem A] that ifA is stable inG (a property which holds whenA or [G:A] is finite), then the setH={g ε G| [A:A ∪A g] εD 1 ∪ {1}} is a subgroup ofG. In this case we say thatA<H<G is a disconnected system (briefly: a system). In the current paper we deal with some fundamental types of systems. A systemA<H<G is irreducible if there does not exist 1<N△G such thatAN<H andAN/N<H/N<G/N is a system. Theorem A gives restrictions on the finite nilpotent normal subgroups ofG, whenG possesses an irreducible system. In particular, ifG is finite then Fit(G) is aq-group for a certain primeq. We deal also with general systems. Corollary (4.2) gives information about the structure of a finite groupG which possesses a system. Theorem B says that for any systemA<H<G,N G (N G (A))=N G (A). Theorem C and Corollary C’ generalize a result of Praeger [P, Theorem 2].
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References
[BL] G. M. Bergman and H. W. Lenstra, Jr.,Subgroups close to normal subgroups, Journal of Algebra127 (1989), 80–97.
[IP] I. M. Isaacs and C. E. Praeger,Permutation group subdegrees and the common divisor graph, Journal of Algebra159 (1993), 158–175.
[K] G. Kaplan,On groups admitting a disconnected common divisor graph, Journal of Algebra193 (1997), 616–628.
[P] C. E. Praeger,Subgroups close to normal subgroups: a commentary on a paper of G.M. Bergman and H.W. Lenstra, Jr., Mathematics Research Report, Australian National University, 1989.
[W] H. Wielandt,Finite Permutation Groups, Academic Press, New York, 1964.
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The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at Tel Aviv University under the supervision of Prof. Marcel Herzog.
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Kaplan, G. Irreducible disconnected systems in groups. Isr. J. Math. 111, 203–219 (1999). https://doi.org/10.1007/BF02810685
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DOI: https://doi.org/10.1007/BF02810685