Recognition of characteristically simple group $A_5\times A_5$ by character degree graph and order

Khademi, Maryam   ; Khosravi, Behrooz

Czechoslovak Mathematical Journal, Tome 68 (2018) no. 4, pp. 1149-1157

Abstract (VO)

The character degree graph of a finite group $G$ is the graph whose vertices are the prime divisors of the irreducible character degrees of $G$ and two vertices $p$ and $q$ are joined by an edge if $pq$ divides some irreducible character degree of $G$. It is proved that some simple groups are uniquely determined by their orders and their character degree graphs. But since the character degree graphs of the characteristically simple groups are complete, there are very narrow class of characteristically simple groups which are characterizable by this method. \endgraf We prove that the characteristically simple group $A_5 \times A_5 $ is uniquely determined by its order and its character degree graph. We note that this is the first example of a non simple group which is determined by order and character degree graph. As a consequence of our result we conclude that $A_5\times A_5$ is uniquely determined by its complex group algebra.

Détail

DOI : 10.21136/CMJ.2018.0134-17

Classification : 20C15, 20D05, 20D08, 20D60
Keywords: character degree graph; irreducible character; characteristically simple group; complex group algebra

Bibliographie

[1] Brauer, R.: Representations of finite groups. Lectures on Modern Mathematics, Vol. I Wiley, New York (1963), 133-175. | MR | JFM

[2] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford (1985). | MR | JFM

[3] Dade, E. C.: Deux groupes finis distincts ayant la môme algêbre de groupe sur tout corps. Math. Z. 119 (1971), 345-348 French. | DOI | MR | JFM

[4] Holt, D. F., Plesken, W.: Perfect Groups. Oxford Mathematical Monographs, Clarendon Press, Oxford (1989). | MR | JFM

[5] Huppert, B.: Character Theory of Finite Groups. De Gruyter Expositions in Mathematics 25, Walter de Gruyter, Berlin (1998). | DOI | MR | JFM

[6] Isaacs, I. M.: Character Theory of Finite Groups. Pure and Applied Mathematics 69, Academic Press, New York (1976). | DOI | MR | JFM

[7] Isaacs, I. M.: Finite Group Theory. Graduate Studies in Mathematics 92, American Mathematical Society, Providence (2008). | DOI | MR | JFM

[8] Jones, M. R.: Some inequalities for the multiplicator of finite group. Proc. Am. Math. Soc. 39 (1973), 450-456. | DOI | MR | JFM

[9] Khosravi, B., Khosravi, B., Khosravi, B.: Recognition of $ PSL(2, p)$ by order and some information on its character degrees where $p$ is a prime. Monatsh. Math. 175 (2014), 277-282. | DOI | MR | JFM

[10] Khosravi, B., Khosravi, B., Khosravi, B.: Some extensions of $ PSL(2,p^2)$ are uniquely determined by their complex group algebras. Commun. Algebra 43 (2015), 3330-3341. | DOI | MR | JFM

[11] Khosravi, B., Khosravi, B., Khosravi, B.: A new characterization for some extensions of $ PSL(2,q)$ for some $q$ by some character degrees. Proc. Indian Acad. Sci., Math. Sci. 126 (2016), 49-59. | DOI | MR | JFM

[12] Khosravi, B., Khosravi, B., Khosravi, B., Momen, Z.: Recognition by character degree graph and order of the simple groups of order less than 6000. Miskolc Math. Notes 15 (2014), 537-544. | DOI | MR | JFM

[13] Khosravi, B., Khosravi, B., Khosravi, B., Momen, Z.: A new characterization for the simple group $ PSL(2, p^2)$ by order and some character degrees. Czech. Math. J. 64 (2015), 271-280. | DOI | MR | JFM

[14] Khosravi, B., Khosravi, B., Khosravi, B., Momen, Z.: Recognition of the simple group $ PSL(2,p^2)$ by character degree graph and order. Monatsh. Math. 178 (2015), 251-257. | DOI | MR | JFM

[15] Khosravi, B., Khosravi, B., Khosravi, B., Momen, Z.: Recognition of some simple groups by character degree graph and order. Math. Rep., Buchar. 18 (68) (2016), 51-61. | MR | JFM

[16] Kimmerle, W.: Group rings of finite simple groups. Resen. Inst. Mat. Estat. Univ. São Paulo 5 (2002), 261-278. | MR | JFM

[17] Lewis, M. L.: An overview of graphs associated with character degrees and conjugacy class sizes in finite groups. Rocky Mt. J. Math. 38 (2008), 175-211. | DOI | MR | JFM

[18] Manz, O., Staszewski, R., Willems, W.: On the number of components of a graph related to character degrees. Proc. Am. Math. Soc. 103 (1988), 31-37. | DOI | MR | JFM

[19] Nagl, M.: Über das Isomorphieproblem von Gruppenalgebren endlicher einfacher Gruppen. Diplomarbeit. Universität Stuttgart (2008), German.

[20] Nagl, M.: Charakterisierung der Symmetrischen Gruppen durch ihre komplexe Gruppenalgebra. Stuttgarter Mathematische Berichte 2011 Universität Stuttgart. Fachbereich Mathematik, Stuttgart (2011), 18, Preprint ID 2011-007. Avaible at http://www.mathematik.uni-stuttgart.de/preprints/downloads/2011/2011-007.pdf German.

[21] Tong-Viet, H. P.: Symmetric groups are determined by their character degrees. J. Algebra 334 (2011), 275-284. | DOI | MR | JFM

[22] Tong-Viet, H. P.: Alternating and sporadic simple groups are determined by their character degrees. Algebr. Represent. Theory 15 (2012), 379-389. | DOI | MR | JFM

[23] Tong-Viet, H. P.: Simple classical groups of Lie type are determined by their character degrees. J. Algebra 357 (2012), 61-68. | DOI | MR | JFM

[24] Tong-Viet, H. P.: Simple exceptional groups of Lie type are determined by their character degrees. Monatsh. Math. 166 (2012), 559-577. | DOI | MR | JFM

[25] White, D. L.: Degree graphs of simple groups. Rocky Mt. J. Math. 39 (2009), 1713-1739. | DOI | MR | JFM

[26] Xu, H., Chen, G., Yan, Y.: A new characterization of simple $K_3$-groups by their orders and large degrees of their irreducible characters. Commun. Algebra 42 (2014), 5374-5380. | DOI | MR | JFM

[27] Xu, H., Yan, Y., Chen, G.: A new characterization of Mathieu-groups by the order and one irreducible character degree. J. Inequal. Appl. Paper No. 209 (2013), 6 pages \99999DOI99999 10.1186/1029-242X-2013-209 \goodbreak. | MR | JFM