On graphs admitting arc-transitive actions of almost simple groups. (English) Zbl 0913.05051
A finite connected regular graph \(\Gamma\) is said to be \(G\)-locally primitive, \(G\leq \operatorname{Aut}\Gamma\), if for each vertex \(\alpha\) of \(\Gamma\), the stabilizer \(G_\alpha\) acts primitively on the set of neighbors of \(\alpha\). The paper is devoted to the study of non-bipartite finite connected vertex-transitive \(G\)-locally primitive graphs \(\Gamma\) with \(G\) being an almost simple group whose socle \(S\) does not act semiregularly on the vertices of \(\Gamma\) and \(C_{\operatorname{Aut} \Gamma} (S)=1\). The significance of this class of graphs lies in its connection to the theory of arc-transitive quasi-primitive graphs originally introduced by the second of the authors.
In the main result of the paper, it is shown that any graph \(\Gamma\) with the above described properties must satisfy one of the following: (a) \(S \trianglelefteq \operatorname{Aut}\Gamma \leq\operatorname{Aut} (S)\); or (b) \(G<Y\leq \operatorname{Aut}\Gamma\), where \(Y\) is almost simple with \(\text{soc} Y\neq S\) and either \(N_Y(G)\) is maximal in \(Y\), or \(\Gamma\) is the complete graph \(K_8\) and \(G=\text{PSL} (2,7)\), \(Y=A_8\) or \(S_8\); or (c) \(\operatorname{Aut}\Gamma\) contains a subgroup \(N\cdot G\), where \(N={\mathcal Z}^d_p\) for some prime \(p\) and \(d>1\), and \(S=S(q)\) is a Lie type simple group over a field of order \(q=p^e\), \(e| d\). Additional restrictions are presented under which case (b) of the main theorem may not occur. For connected \((G,2)\)-arc-transitive graphs with \(\text{Sz} (q)\leq G\leq \operatorname{Aut} (\text{Sz} (q))\) \((q=2^{2n+1} \geq 8)\) or \(G=\text{Ree} (q)\) \((q=3^{2n+1} \geq 27)\) it is shown that \(G\leq\operatorname{Aut} \Gamma \leq \operatorname{Aut} (G)\).
In the main result of the paper, it is shown that any graph \(\Gamma\) with the above described properties must satisfy one of the following: (a) \(S \trianglelefteq \operatorname{Aut}\Gamma \leq\operatorname{Aut} (S)\); or (b) \(G<Y\leq \operatorname{Aut}\Gamma\), where \(Y\) is almost simple with \(\text{soc} Y\neq S\) and either \(N_Y(G)\) is maximal in \(Y\), or \(\Gamma\) is the complete graph \(K_8\) and \(G=\text{PSL} (2,7)\), \(Y=A_8\) or \(S_8\); or (c) \(\operatorname{Aut}\Gamma\) contains a subgroup \(N\cdot G\), where \(N={\mathcal Z}^d_p\) for some prime \(p\) and \(d>1\), and \(S=S(q)\) is a Lie type simple group over a field of order \(q=p^e\), \(e| d\). Additional restrictions are presented under which case (b) of the main theorem may not occur. For connected \((G,2)\)-arc-transitive graphs with \(\text{Sz} (q)\leq G\leq \operatorname{Aut} (\text{Sz} (q))\) \((q=2^{2n+1} \geq 8)\) or \(G=\text{Ree} (q)\) \((q=3^{2n+1} \geq 27)\) it is shown that \(G\leq\operatorname{Aut} \Gamma \leq \operatorname{Aut} (G)\).
Reviewer: R.Jajcay (Terre Haute)
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
References:
| [1] | R. W. Baddeley, C. E. Praeger, On Primitive Overgroups of Quasiprimitive Permutation Groups, University of Leicester, 1996; R. W. Baddeley, C. E. Praeger, On Primitive Overgroups of Quasiprimitive Permutation Groups, University of Leicester, 1996 · Zbl 1024.20002 |
| [2] | Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups (1985), Clarendon Press: Clarendon Press Oxford · Zbl 0568.20001 |
| [3] | X. G. Fang, C. E. Praeger, Finite Two Arc Transitive Graphs Admitting a Suzuki Simple Group, Comm. Algebra; X. G. Fang, C. E. Praeger, Finite Two Arc Transitive Graphs Admitting a Suzuki Simple Group, Comm. Algebra · Zbl 0956.05049 |
| [4] | X. G. Fang, C. E. Praeger, Finite Two Arc Transitive Graphs Admitting a Ree Simple Group, Comm. Algebra; X. G. Fang, C. E. Praeger, Finite Two Arc Transitive Graphs Admitting a Ree Simple Group, Comm. Algebra · Zbl 0945.05031 |
| [5] | Hall, M., The Theory of Groups (1959), Macmillan: Macmillan New York · Zbl 0084.02202 |
| [6] | Issacs, M., Character Theory of Finite Groups (1976), Academic Press: Academic Press New York · Zbl 0337.20005 |
| [7] | James, G. D., On the minimal dimensions of irreducible representations of symmetric groups, Math. Proc. Cambridge Philos. Soc., 94, 417-424 (1983) · Zbl 0544.20011 |
| [8] | Jansen, C.; Lux, K.; Parker, R.; Wilson, R., An Atlas of Brauer Characters (1995), Clarendon Press: Clarendon Press Oxford · Zbl 0831.20001 |
| [9] | Kantor, W. M., \(k\), Math. Z., 124, 274-292 (1972) |
| [10] | Kleidman, P.; Liebeck, M., The Subgroup Structure of the Finite Classical Groups (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0697.20004 |
| [11] | Landazuri, V.; Seitz, G. M., On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra, 32, 418-473 (1974) · Zbl 0325.20008 |
| [12] | Liebeck, M. W., On the orders of maximal subgroups of the finite classic groups, Proc. London Math. Soc. (3), 50, 426-446 (1985) · Zbl 0591.20021 |
| [13] | Liebeck, M. W.; Praeger, C. E.; Saxl, J., The maximal factorizations of the finite simple groups and their automorphism groups, Mem. Amer. Math. Soc., 86 (1990) · Zbl 0703.20021 |
| [14] | Libeck, M. W.; Praeger, C. E.; Saxl, J., On factorizations of almost simple groups, J. Algebra, 185, 409-419 (1996) · Zbl 0862.20016 |
| [15] | M. W. Liebeck, C. E. Praeger, J. Saxl, On the automorphism groups of finite vertex-primitive, edge-transitive graphs and directed graphs; M. W. Liebeck, C. E. Praeger, J. Saxl, On the automorphism groups of finite vertex-primitive, edge-transitive graphs and directed graphs · Zbl 0703.20021 |
| [16] | Praeger, C. E., Imprimitive symmetric graphs, Ars Combin., 19A, 149-163 (1985) · Zbl 0575.05047 |
| [17] | Praeger, C. E., The inclusion problem for finite primitive permutation groups, Proc. London Math. Soc. (3), 60, 68-88 (1990) · Zbl 0653.20005 |
| [18] | Praeger, C. E., On the O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc. (2), 47, 227-239 (1993) · Zbl 0738.05046 |
| [19] | Ree, R., A family of simple groups associated with the simple Lie algebra of type \((G_2\), Amer. J. Math, 83, 432-462 (1961) · Zbl 0104.24705 |
| [20] | Sabidussi, G. O., Vertex-transitive graphs, Monatsh. Math, 68, 426-438 (1964) · Zbl 0136.44608 |
| [21] | Suzuki, M., On a class of doubly transitive groups, Ann. of Math., 75, 105-145 (1962) · Zbl 0106.24702 |
| [22] | Wagner, A., The faithful linear representations of least degree of\(S_n\)and\(A_n\)over a field of characteristic 2, Math. Z., 151, 127-137 (1976) · Zbl 0321.20008 |
| [23] | Wagner, A., The faithful linear representations of least degree of\(S_n\)and\(A_n\)over a field of odd characteristic, Math. Z., 154, 103-114 (1977) · Zbl 0336.20008 |
| [24] | Wielandt, H., Finite Permutation Groups (1964), Academic Press: Academic Press New York · Zbl 0138.02501 |
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