In this dissertation we completely determine the maximal subgroups of the following finite simple groups: (i) POgX?) and 3D^q) for all prime powers q (ii) 2G2(32m+1) for all integers m (iii) G2(<7) for all odd prime powers q. Moreover, if Go is one of the groups appearing in (i), (ii) or (iii), then we also determine the maximal subgroups of all groups G satisfying: GO<G< Aut{Go\ (*) where Aut{Go) is the automorphism group of Go. Chapter 1 is devoted to the case Go = PClt(.q), where q = pt and p is prime. We first analyse the structure of the full automorphism group A = Aut(Go), as follows. Let Q be a quadratic form of Witt defect O defined on an 8-dimensional vector space V over F = GF(q). We write 0 = 0 (V,F£) for the isometry group of Q. We then define a chain of groups 0 <. SO < O < A < T all related to the geometry (V,¥,Q). The group T is the full semilinear group associated with Q and fl = [0,0] is a perfect group. Upon factoring out scalars, we obtain the projective groups PCI < PSO < PO < PA < PI\ We have Ptl = Go and | A:PT \ = 3. In fact, A is generated by Pr and a triality automorphism, which occurs because the Dynkin diagram of Go admits a symmetry of order 3. We then show that AlGo — Ex Z/, where E is the symmetric group S3 or S4. We thus obtain a homomorphism JT : A —» E whose kernel is isomorphic to GoXf. It turns out that G (as in (*)) contains a triality automorphism if and only if 3 divides | r(G)\. A recent theorem of M. Aschbacher [Invent, meth. 76 (1984), 469-514] shows that if G < PV, then the maximal subgroups of G fall into two families, which we may call C and S. Groups in C can be read off from from Aschbacher's paper, and we determine the groups in S by studying the p- modular representations of the finite simple groups. Thus we appeal to the classification of the finite simple groups. We then consider the case in which G •%. PY. Here G contains a triality automorphism and our argument goes roughly like this. Take Af to be a maximal subgroup of G which satisfies MGO = G and write M o = M n Go. Then M o < L < Go for some maximal subgroup L of Go. But M contains a triality automorphism T and so M o < L n U n Lr2. Now L is known because we have already handled the case in which G < PT (in particular, the case G = Go). Therefore our knowledge of L together with our knowledge concerning the action of r allows us to determine all possibilities for Mo. Hence M is known, for M £- MO.(G/GO). In Chapter 2 we treat the case Go = aD^(q). The group 3D4(<7) is the centralizer in PO^O?3) of a suitable triality automorphism. Thus the information about triauty which we collect in Chapter 1 is exploited in Chapter 2 to obtain the maximal subgroups of 3D^(q) and it automorphism groups. Similarly, G2O7) is the centralizer in PCl^iq) of a suitable triality. Thus in Chapter 3 we deal with the case Go = G2(?) (with q odd) by exploiting triality once again. Our methods for analysing G2O7) readily lend themselves to handle Go = 2Gi{q\ and this work is presented in Chapter 4. Chapter 4 also contains information about the maximal subgroups of the automorphism groups of the Suzuki groups Sz(q) = ^i^fa)- Note that in his original paper, Suzuki find the subgroups of the simple group We however find the maximal subgroups of all groups G satisfying < G < Aut(Sz(q)). In Chapter 5 we present lists of maximal subgroups of several families of low dimensional finite classical groups, including PSLn(q) for 2 < n < 11. We do not include proofs, although we sketch a proof for PSL&(q). Some of these results have appeared much earlier in the literature (dating as far back as the 19th century), but most of them are new.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:234996 |
| Date | January 1987 |
| Creators | Kleidman, Peter Brown |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.repository.cam.ac.uk/handle/1810/250910 |
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