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 Trudy Inst. Mat. i Mekh. UrO RAN, 2015, Volume 21, Number 3, Pages 222–232 (Mi timm1215)

On the finite prime spectrum minimal groups

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 set of all prime divisors of the order of $G$ is called the prime spectrum of $G$ and is denoted by $\pi(G)$. A group $G$ is called prime spectrum minimal if $\pi(G) \not = \pi(H)$ for any proper subgroup$H$ of$G$. We prove that every prime spectrum minimal group all whose non-abelian composition factors are isomorphic to the groups from the set $\{PSL_2(7), PSL_2(11), PSL_5(2)\}$ is generated by two conjugate elements. Thus, we expand the correspondent result for finite groups with Hall maximal subgroups. Moreover, we study the normal structure of a finite prime spectrum minimal group which has a simple non-abelian composition factor whose order is divisible by $3$ different primes only.

Keywords: finite group, generation by a pair of conjugate elements, prime spectrum, prime spectrum minimal group, maximal subgroup, composition factor.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2016, 295, suppl. 1, 109–119

Bibliographic databases:

UDC: 512.542

Citation: N. V. Maslova, “On the finite prime spectrum minimal groups”, Trudy Inst. Mat. i Mekh. UrO RAN, 21, no. 3, 2015, 222–232; Proc. Steklov Inst. Math. (Suppl.), 295, suppl. 1 (2016), 109–119

Citation in format AMSBIB
\Bibitem{Mas15} \by N.~V.~Maslova \paper On the finite prime spectrum minimal groups \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2015 \vol 21 \issue 3 \pages 222--232 \mathnet{http://mi.mathnet.ru/timm1215} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3468106} \elib{http://elibrary.ru/item.asp?id=24156725} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2016 \vol 295 \issue , suppl. 1 \pages 109--119 \crossref{https://doi.org/10.1134/S0081543816090121} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000394441400012}