A characterization of some simple unitary groups via order and degree pattern of solvable graph

Document Type : Research Paper

Author

Department of Mathematics, Sahand University of Technology, Tabriz, IRAN.

10.29252/as.2019.1614

Abstract

The solvable graph associated with a finite group $G$, denoted by ${\Gamma}_{\rm s}(G)$, is a simple graph whose vertices are the prime divisors of $|G|$ and two distinct primes $p$ and $q$ are joined by an edge if and only if there exists a solvable subgroup of $G$ whose order is divisible by $pq$. In this paper, we give a
characterization for projective special unitary groups $U_3(q)$ with some certain conditions by the solvable graph.

Keywords


[1] S. Abe, A characterization of some nite simple groups by orders of their solvable subgroups, Hokkaido Math. J., Vol. 31 (2002), pp. 349-361.
[2] S. Abe and N. Iiyori, A generalization of prime graphs of nite groups, Hokkaido Math. J., Vol. 29 (2000), pp. 391-407.
[3] B. Akbari, N. Iiyori and A. R. Moghaddamfar, A new characterization of some simple groups by order and degree pattern of solvable graph, Hokkaido Math. J., Vol. 45 (2016), pp. 337-363.
[4] B. Akbari, ODs-characterization of some low-dimensional nite classical groups, International Electronic J. Algebra, Vol. 24 (2018), pp. 73-90.
[5] R. Brandl and W. J. Shi, A characterization of nite simple groups with abelian Sylow 2-subgroups, Ricerche Mat., Vol. 42 No. 1 (1993), pp. 193-198.
[6] J. Bray, D. Holt and C. Roney-Dougal, The maximal subgroups of the low-dimensional nite classical groups, Cambridge University Press, Cambridge, (2013).
[7] A. A. Buturlakin, Spectra of nite symplectic and orthogonal groups, Siberian Adv. Math., Vol. 21 No. 3 (2011), pp. 176-210
[8] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, (1985).
[9] P. B. Kleidman, The maximal subgroups of the Chevalley groups G2(q) with q odd, the Ree groups 2G2(q), and their automorphism groups, J. Algebra, Vol. 117 No. 1 (1988), pp. 30-71.
[10] M. S. Lucido, Groups in which the prime graph is a tree, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), Vol. 5 No. 1 (2002), pp. 131-148.
[11] A. V. Zavarnitsine and V. D. Mazurov, Element orders in coverings of symmetric and alternating groups, Algebra Logic, Vol. 38 No. 3 (1999), pp. 159-170.