Abstract
Let G be a finite group and π(G) be the set of all prime divisors of its order. The prime graph GK(G) of G is a simple graph with vertex set π(G), and two distinct primes p, q ∈ π(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p ∈ π(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If π(G) = {p 1, p 2,...,p k }, where p 1 < p 2 < ... < p k , then the degree pattern of G is defined by D(G) = (deg(p 1), deg(p 2),...,deg(p k )). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |H| = |G| and D(H) = D(G). In addition, a 1-fold OD-characterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A 22 is OD-characterizable. We also show that the automorphism groups of the alternating groups A 16 and A 22, i.e., the symmetric groups S 16 and S 22 are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected.
Similar content being viewed by others
References
Aschbacher M. Finite Group Theory. Cambridge: Cambridge University Press, 1986
Carter R W. Simple Groups of Lie type. Pure and Applied Mathematics, Vol 28. London-New York-Sydney: John Wiley and Sons, 1972
Conway J H, Curtis R T, Norton S P, Parker R A, Wilson R A. Atlas of Finite Groups. Oxford: Clarendon Press, 1985
Kleidman P, Liebeck M. The Subgroup Structure of the Finite Classical Groups. London Mathematical Society Lecture Note Series, 129. Cambridge: Cambridge University Press, 1990
Kondratév A S. On prime graph components of finite simple groups. Math Sb, 1989, 180(6): 787–797
Lucido M S. Prime graph components of finite almost simple groups. Rend Sem Mat Univ Padova, 1999, 102: 1–22
Moghaddamfar A R, Rahbariyan S. More on the OD-characterizability of a finite group. Algebra Colloq (to appear)
Moghaddamfar A R, Zokayi A R. Recognizing finite groups through order and degree pattern. Algebra Colloq, 2008, 15(3): 449–456
Moghaddamfar A R, Zokayi A R. OD-Characterization of certain finite groups having connected prime graphs. Algebra Colloq (to appear)
Moghaddamfar A R, Zokayi A R, Darafsheh M R. A characterization of finite simple groups by the degrees of vertices of their prime graphs. Algebra Colloq, 2005, 12(3): 431–442
Robinson D J S. A Course in the Theory of Groups. New York: Springer-Verlag, 1982
Williams J S. Prime graph components of finite groups. J Algebra, 1981, 69(2): 487–513
Zavarnitsin A. Finite simple groups with narrow prime spectrum. Siberian Electronic Mathematical Reports, 2009, 6: 1–12
Zavarnitsin A, Mazurov V D. Element orders in covering of symmetric and alternating groups. Algebra and Logic, 1999, 38(3): 159–170
Zhang L C, Shi W J. OD-Characterization of all simple groups whose orders are less than 108. Front Math China, 2008, 3(3): 461–474
Zhang L C, Shi WJ. OD-Characterization of almost simple groups related to L2(49). Arch Math (Brno), 2008, 44(3): 191–199
Zhang L C, Shi W J. OD-Characterization of simple K 4-groups. Algebra Colloq, 2009, 16(2): 275–282
Zhang L C, Shi W J, Wang L L, Shao C G. OD-Characterization of A16. Journal of Suzhou University (Natural Science Edition), 2008, 24(2): 7–10
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Moghaddamfar, A.R., Zokayi, A.R. OD-Characterization of alternating and symmetric groups of degrees 16 and 22. Front. Math. China 4, 669–680 (2009). https://doi.org/10.1007/s11464-009-0037-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-009-0037-1