Finite simple groups that are not spectrum critical

N. V. Maslova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 172-176

Résumé

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 $G$.

Détail

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

Bibliographie

[1] Buturlakin A.A., “Spektry konechnykh simplekticheskikh i ortogonalnykh grupp”, Mat. trudy, 13:2 (2010), 33–83 | MR | Zbl

[2] Buturlakin A.A., “Spektry konechnykh lineinykh i unitarnykh grupp”, Algebra i logika, 47:2 (2008), 157–173 | MR | Zbl

[3] Vasilev A.V., Grechkoseeva M.A., Mazurov V.D., “O konechnykh gruppakh, izospektralnykh prostym simplekticheskim i ortogonalnym gruppam”, Sib. matem. zhurn., 50:6 (2009), 1225–1247 | MR | Zbl

[4] Gorshkov I.B., “Raspoznavaemost znakoperemennykh grupp po spektru”, Algebra i logika, 52:1 (2013), 57–63 | MR | Zbl

[5] Zavarnitsin A.V., “Raspoznavanie prostykh grupp $U_3(q)$ po poryadkam elementov”, Algebra i logika, 45:2 (2006), 185–202 | MR | Zbl

[6] Mazurov V.D., Shi V., “Priznak neraspoznavaemosti konechnoi gruppy po spektru”, Algebra i logika, 51:2 (2012), 239–243 | MR | Zbl

[7] Maslova N.V., “O sovpadenii grafov Gryunberga – Kegelya konechnoi prostoi gruppy i ee sobstvennoi podgruppy”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:1 (2014), 156–168 | MR

[8] Maslova N.V., Revin D.O., “Porozhdaemost konechnoi gruppy s khollovymi maksimalnymi podgruppami paroi sopryazhennykh elementov”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19:3 (2013), 199–206 | MR

[9] Maslova N.V., Revin D.O., “O neabelevykh kompozitsionnykh faktorakh konechnoi gruppy, minimalnoi otnositelno prostogo spektra”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19:4 (2013), 155–166 | MR

[10] Nereshennye voprosy teorii grupp. Kourovskaya tetrad, Izd. 17-e, dop., In-t matematiki SO RAN, Novosibirsk, 2010, 219 pp.

[11] Staroletov A.M., “O raspoznavanii nekotorykh prostykh ortogonalnykh grupp po spektru”, Algebra i kombinatorika, tez. mezhdunar. konf. po algebre i kombinatorike, posvyasch. 60-letiyu A.A. Makhneva, izd-vo “UMTs-UPI”, Ekaterinburg, 2013, 144

[12] Aschbacher M., Finite group theory, Cambridge Univ. Press, Cambridge, 1986, 274 pp. | MR | Zbl

[13] J.H. Conway [et. al.], Atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp. | MR | Zbl

[14] Bray J.N., Holt D.F., Roney-Dougal C.M., The maximal subgroups of the low-dimensional finite classical groups, Cambridge Univ. Press, Cambridge, 2013, 438 pp. | MR | Zbl

[15] Buturlakin A.A., “Isospectral finite simple groups”, Sib. elektron. mat. izv., 7 (2010), 111–114 | MR

[16] Gorenstein D., Finite groups, Chelsea Publishing Company, New York, 1968, 519 pp. | MR

[17] Gorenstein D., Lyons R., Solomon R., The classification of the finite simple groups, Number 3, Math. Surveys Monogr., 40.3, Providence, 1998, 419 pp. | MR | Zbl

[18] Liebeck M.W., Praeger C.E., Saxl J., “Transitive subgroups of primitive permutation groups”, J. Algebra, 234:2 (2000), 291–361 | DOI | MR | Zbl

[19] M.A. Grechkoseeva, M.S. Lucido, V.D. Mazurov, A.R. Moghaddamfar, A.V. Vasilev, “On recognition of the projective special linear groups over binary fields”, Sib. elektron. mat. izv., 2 (2005), 253–263 | MR | Zbl