Finite simple groups that are not spectrum critical
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 172-176
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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.
@article{TIMM_2015_21_1_a16,
author = {N. V. Maslova},
title = {Finite simple groups that are not spectrum critical},
journal = {Trudy Instituta matematiki i mehaniki},
pages = {172--176},
publisher = {mathdoc},
volume = {21},
number = {1},
year = {2015},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a16/}
}
N. V. Maslova. Finite simple groups that are not spectrum critical. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 172-176. http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a16/