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 Trudy Inst. Mat. i Mekh. UrO RAN, 2015, Volume 21, Number 1, Pages 172–176 (Mi timm1153)

Finite simple groups that are not spectrum critical

N. V. Maslovaab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg

Abstract: Let $G$ be a finite group. The spectrum of $G$ is the set $\omega(G)$ of orders of all its elements. The subset of prime elements of $\omega(G)$ is called prime spectrum and is denoted by $\pi(G)$. A group $G$ is called spectrum critical ( prime spectrum critical) if, for any subgroups $K$ and $L$ of $G$ such that $K$ is a normal subgroup of $L$, the equality $\omega(L/K)=\omega(G)$ ($\pi(L/K)=\pi(G)$, respectively) implies that $L=G$ and $K=1$. In the present paper, we describe all finite simple groups that are not spectrum critical. In addition, we show that a prime spectrum minimal group $G$ is prime spectrum critical if and only if its Fitting subgroup $F(G)$ is a Hall subgroup of $G$.

Keywords: finite group; simple group; spectrum; prime spectrum; spectrum critical group; prime spectrum critical group.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2016, 292, suppl. 1, 211–215

Bibliographic databases:

UDC: 512.542

Citation: N. V. Maslova, “Finite simple groups that are not spectrum critical”, Trudy Inst. Mat. i Mekh. UrO RAN, 21, no. 1, 2015, 172–176; Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 211–215

Citation in format AMSBIB
\Bibitem{Mas15} \by N.~V.~Maslova \paper Finite simple groups that are not spectrum critical \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2015 \vol 21 \issue 1 \pages 172--176 \mathnet{http://mi.mathnet.ru/timm1153} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3379614} \elib{http://elibrary.ru/item.asp?id=23137985} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2016 \vol 292 \issue , suppl. 1 \pages 211--215 \crossref{https://doi.org/10.1134/S0081543816020176} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000376272600017} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84971572342} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. V. Maslova, “Finite groups with arithmetic restrictions on maximal subgroups”, Algebra and Logic, 54:1 (2015), 65–69
2. A. Pachera, “Exponent preserving subgroups of the finite simple groups”, Commun. Algebr., 45:6 (2017), 2494–2504
3. Yuri V. Lytkin, “On finite groups isospectral to the simple groups $S_4(q)$”, Sib. elektron. matem. izv., 15 (2018), 570–584
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