Intersections of Nilpotent Subgroups in Finite Groups with Sporadic Socle
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Algebra and Logic, Vol. 59, No. 4, September, 2020 (Russian Original Vol. 59, No. 4, July-August, 2020)
INTERSECTIONS OF NILPOTENT SUBGROUPS IN FINITE GROUPS WITH SPORADIC SOCLE V. I. Zenkov∗
UDC 512:542
Keywords: sporadic group, nilpotent subgroup, intersections of subgroups. It is proved that for any nilpotent subgroups A and B in a finite group G with sporadic
socle, there is an element g such that A B g = 1.
INTRODUCTION Let G be a finite group and A and B be subgroups of G. We define MG (A, B) to be a set of
minimal (w.r.t. inclusion) intersections of the form A B g , g ∈ G, and define mG (A, B) to be a set of minimal (w.r.t. the order) elements of MG (A, B). Then mG (A, B) ⊆ MG (A, B) by definition. Set MinG (A, B) = MG (A, B) and minG (A, B) = mG (A, B). It is clear that MinG (A, B) ≥ minG (A, B). There are examples, both for soluble and almost simple groups, where MinG (A, B) > minG (A, B) and the subgroup MinG (A, B) may not be in F (G) while the subgroup minG (A, B) is in F (G). For instance, in a group G = Σ4 , for its subgroups we have E4 B < A D8 . If B = O2 (G), then minG (A, B) = O2 (G) = MinG (A, B). And if B O2 (G), then minG (A, B) = O2 (G) < MinG (A, B) = A. Note that in this case one of the subgroups is Abelian and the other is minimal non-Abelian. However, if, in an arbitrary finite group G, both subgroups A and B are Abelian, then MinG (A, B) ≤ F (G), where F (G) is a Fitting subgroup of G [1, Thm. 1]. Even for a group G = Aut (L3 (2)) with A = B ∈ Syl2 (G), we have MinG (A, B) = A = 1 [2, Thm. 2]. For the case where the subgroups A and B of a simple non-Abelian group G are primary, it was proved in [3] that MinG (A, B) = 1. ∗
Supported by RFBR (project No. 20-01-00456) and by the Competitiveness Enhancement Program for leading universities of Russia (Agreement No. 02.A03.21.0006 of 27.08.2013).
Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences. El’tsyn Ural Federal University, Yekaterinburg, Russia; [email protected]. Translated from Algebra i Logika, Vol. 59, No. 4, pp. 458-470, July-August, 2020. Russian DOI: 10.33048/alglog.2020.59.403. Original article submitted March 20, 2020; accepted November 24, 2020. c 2020 Springer Science+Business Media, LLC 0002-5232/20/5904-0313
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The objective of the present paper is to study intersections of two nilpotent subgroups of a finite group with a sporadic socle. 1. NOTATION AND PRELIMINARY RESULTS The notation used in the paper is basically standard. For part of it, see [4, 5]. LEMMA 1.1 [1, Thm. 1]. Let G be a finite group and A and B be Abelian subgroups of G. Then minG (A, B) ≤ MinG (A, B) ≤ F (G). LEMMA 1.2 [6, Lemma 2.1]. Let G be a finite group, A be a cyclic subgroup of G, and B be a nilpotent subgroup of G. Then minG (A, B) ≤ MinG (A, B) ≤ F (G). LEMMA 1.3 [3, Thm. 1]. Let G be a finite simple non-Abelian group and A and B be primary subgroups of G. Then MinG (A, B) = minG (A, B) = 1. LEMMA 1.4 [7, Thm. 2]. Let G be a finite insoluble group with a socle isomorphic to L2 (q), A and B be n
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