Geodesic
On recognition of $A_6\times A_6$ by the set of conjugacy class sizes
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 762-767 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For a finite group $G$ denote by $N(G)$ the set of conjugacy class sizes of $G$. Recently the following question has been asked: Is it true that for each nonabelian finite simple group $S$ and each $n\in\mathbb{N}$, if the set of class sizes of a finite group $G$ with trivial center is the same as the set of class sizes of the direct power $S^n$, then $G\simeq S^n$? In this paper we approach an answer to this question by proving that $A_6\times A_6$ is uniquely determined by $N(A_6\times A_6)$ among finite groups with trivial center.
Keywords: finite groups, class sizes.
Mots-clés : conjugacy classes 
@article{SEMR_2022_19_2_a4,
     author = {V. Panshin},
     title = {On recognition of~$A_6\times A_6$ by the set of conjugacy class sizes},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {762--767},
     year = {2022},
     volume = {19},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a4/}
}
TY  - JOUR
AU  - V. Panshin
TI  - On recognition of $A_6\times A_6$ by the set of conjugacy class sizes
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2022
SP  - 762
EP  - 767
VL  - 19
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a4/
LA  - en
ID  - SEMR_2022_19_2_a4
ER  - 
%0 Journal Article
%A V. Panshin
%T On recognition of $A_6\times A_6$ by the set of conjugacy class sizes
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2022
%P 762-767
%V 19
%N 2
%U http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a4/
%G en
%F SEMR_2022_19_2_a4
V. Panshin. On recognition of $A_6\times A_6$ by the set of conjugacy class sizes. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 762-767. http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a4/

[1] A.R. Camina, “Arithmetical conditions on the conjugacy class numbers of a finite group”, J. Lond. Math. Soc., II. Ser., 5 (1972), 127–132 | DOI | MR | Zbl

[2] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985 | MR | Zbl

[3] D. Gorenstein, Finite groups, Harper Row Publishers, New York etc, 1968 | MR | Zbl

[4] I.B. Gorshkov, “On Thompson's conjecture for finite simple groups”, Commun. Algebra, 47:12 (2019), 5192–5206 | DOI | MR | Zbl

[5] I.B. Gorshkov, “On characterization of a finite group by the set of conjugacy class sizes”, J. Algebra Appl., 21:11 (2022), 2250226 | DOI | MR | Zbl

[6] E.I. Khukhro, V.D. Mazurov (eds.), The Kourovka notebook. Unsolved problems in group theory, 2022, arXiv: 1401.0300 | MR