Geodesic
On groups critical with respect to a set of natural numbers
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 10 (2013), pp. 666-675.

Voir la notice de l'article provenant de la source Math-Net.Ru

The spectrum of a finite group is the set of its element orders. A finite group $G$ is critical with respect to a subset $\omega$ of natural numbers, if $\omega$ is equal to the spectrum of $G$ and not equal to the spectrum of any proper section of $G$. For any natural number $n$, we construct $n$ finite critical groups with the same spectrum. We also give a complete description of finite groups critical with respect to the spectrum of the alternating group of degree 6 and the spectrum of the special linear group of dimension 3 over a field of order 3.
Keywords: finite group, spectrum, critical group. 
@article{SEMR_2013_10_a18,
     author = {Yu. V. Lytkin},
     title = {On groups critical with respect to a set of natural numbers},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {666--675},
     publisher = {mathdoc},
     volume = {10},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2013_10_a18/}
}
TY  - JOUR
AU  - Yu. V. Lytkin
TI  - On groups critical with respect to a set of natural numbers
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2013
SP  - 666
EP  - 675
VL  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2013_10_a18/
LA  - en
ID  - SEMR_2013_10_a18
ER  - 
%0 Journal Article
%A Yu. V. Lytkin
%T On groups critical with respect to a set of natural numbers
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2013
%P 666-675
%V 10
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2013_10_a18/
%G en
%F SEMR_2013_10_a18
Yu. V. Lytkin. On groups critical with respect to a set of natural numbers. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 10 (2013), pp. 666-675. http://geodesic.mathdoc.fr/item/SEMR_2013_10_a18/

[1] V. D. Mazurov, W. J. Shi, “A criterion of unrecignizability by spectrum for finite groups”, Algebra and Logic, 51:2 (2012), 239–243 | MR | Zbl

[2] R. Brandl, W. J. Shi, “Finite groups whose element orders are consecutive integers”, J. Algebra, 143:2 (1991), 388–400 | MR | Zbl

[3] V. D. Mazurov, “Recognition of finite simple groups $S_4(q)$ by their element orders”, Algebra and Logic, 41:2 (2002), 166–198 | MR | Zbl

[4] J. S. Williams, “Prime graph components of finite groups”, J. Algebra, 69:2 (1981), 487–513 | MR | Zbl

[5] J. G. Thompson, “Normal $p$-complements for finite groups”, Math. Z., 72:2 (1960), 332–354 | MR | Zbl

[6] H. Zassenhaus, “Kennzeichnung endlicher linearean Gruppen als Permutationsgruppen”, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 17–40 | MR

[7] H. Zassenhaus, “Über endliche Fastkörper”, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 187–220 | MR

[8] B. Huppert, Endliche Gruppen, Springer-Verlag, Berlin–Heidelberg–New York, 1979

[9] D. Gorenstein, Finite groups, Chelsea Publishing Company, New York, N. Y., 1980 | MR | Zbl

[10] A. V. Zavarnitsine, “Finite simple groups with narrow prime spectrum”, Siberian electronic mathematical reports, 6 (2009), 1–12 | MR

[11] M. R. Aleeva, “On finite simple groups with the set of element orders as in a Frobenius group or a double Frobenius group”, Mathematical Notes, 73:3 (2003), 299–313 | MR | Zbl

[12] V. D. Mazurov, “A generalization of a theorem of Zassenhaus”, Vladikavkaz. Mat. Zh., 10:1 (2008), 40–52 | MR

[13] D. V. Lytkina, “Structure of a group with elements of order at most 4”, Siberian Math. J., 48:2 (2007), 283–287 | MR | Zbl

[14] GAP: Groups, algorithms, and programming http://www.gap-system.org