On subgroups containing non-trivial normal subgroups |
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| Authors: | Marcel Herzog Gil Kaplan Andrea Lucchini |
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| Institution: | (1) School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978 Tel Aviv, Israel;(2) Department of Computer Sciences, The Academic College of Tel Aviv-Yaffo, 4 Antokolsky St., 64044 Tel Aviv, Israel;(3) Dipartimento di Elettronica per l’Automazione, Univesità degli Studi di Brescia, Via Branze, 25123 Brescia, Italy |
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| Abstract: | We prove that ifA≠1 is a subgroup of a finite groupG and the order of an element in the centralizer ofA inG is strictly larger (larger or equal) than the index G:A], thenA contains a non-trivial characteristic (normal) subgroup ofG. Consequently, ifA is a stabilizer in a transitive permutation group of degreem>1, thenexp(Z(A))<m. These theorems generalize some recent results of Isaacs and the authors. |
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