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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.

We briefly discuss the algorithm given in [Bates, Bundy, Perkins, Rowley, J. Algebra, 316(2):849-868, 2007] for determining the distance between two vertices in a commuting involution graph of a symmetric group.We develop the algorithm in [Bates, Rowley, Arch. Math. (Basel), 85(6):485-489, 2005] for computing a subgroup of the normalizer of a 2-subgroup X in a finite group G, examining in particular the issue of when to terminate the randomized procedure. The resultant algorithm is capable of handling subgroups X of order up to 512 and is suitable, for example, for matrix groups of large degree (an example calculation is given using 112x112 matrices over GF(2)).We also determine the suborbits of conjugacy classes of involutions in several of the sporadic simple groups?namely Janko's group J4, the Fischer sporadic groups, and the Thompson and Harada-Norton groups. We use our results to determine the structure of some graphs related to this data.We include implementations of the algorithms discussed in the computer algebra package MAGMA, as well as representative elements for the involution suborbits.

Our aim is to calculate some graphs associated with two of the larger sporadicsimple groups, Fi₂₄ and the Baby Monster. Firstly we calculate the point line collinearity graph for a maximal 2-local geometry of Fi₂₄. If T is such a geometry, then the point line collinearity graph G will be the graph whose vertices are the points in T, with any two vertices joined by an edge if and only if they are incident with a common line. We found that the graph has diameter 5 and we give its collapsed adjacency matrix. We also calculate part of the commuting involution graph, C, for the class 2C of the Baby Monster, whose vertex set is the conjugacy class 2C, with any two elements joined by an edge if and only if they commute. We have managed to place all vertices inside C whose product with a fixed vertex t does not have 2 power order, with all evidence pointing towards C having diameter 3.

In this thesis, we will see a way to use representation theory and the theory of linear algebraic groups to characterize certain family of finite groups. In Chapter 1, we see the history of preceding work. In particular, J. G. Thompson’s classification of minimal finite simple nonsolvable groups and characterization of solvable groups will be given. In Chapter 2, we will describe some background knowledge underlying this project and notation that will be widely used in this thesis. In Chapter 3, the main theorem originally conjectured by Guralnick and Tiep will be stated together with the base theorem which is a reduced version of main theorem to the case where we have a quasisimple group. Main theorem explains a way to characterize the finite groups with a composition factor of order divisible by two distinct primes p and q as the finite groups containing nontrivial 2-element x, p-element y, q-element z such that xyz = 1. In this thesis we more focus on the proof of showing a finite group G with a composition factor of order divisible by two distinct prime p and q contains nontrivial 2-element x, p-element y, q-element z such that xyz = 1. In Chapter 4, we will prove a set of lemmas and proposition which will be used as key tools in the proof of the base theorem. In Chapters 5 to 7, we will establish the base theorem in the cases where a quasisimple group G has its simple quotient isomorphic to alternating groups or sporadic groups (Chapter 5), classical groups (Chapter 6), and exceptional groups (Chapter 7). In Chapter 8, we show that any finite group G admitting nontrivial 2-element x, p- element y, q-element z such that xyz = 1 for two distinct odd primes p and q admits a composition factor of order divisible by pq. Also, we show that the question if a finite group G with a composition factor of order divisible by two distinct prime p and q contains nontrivial 2-element x, p-element y, q-element z such that xyz = 1 can be reduced to the base theorem.

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In this thesis, we mainly address two contrasting topics in the area of saturated fusion systems. The first concerns the notion of a centraliser of a subsystem E of a fusion system F, and we give new proofs of the existence of such an object in the case where E is normal in F. The second concerns the development of the theory of `trees of fusion systems', an analogue for fusion systems of Bass-Serre theory for finite groups. A major theorem finds conditions on a tree of fusion systems for there to exist a saturated completion, and this is applied to construct and classify certain fusion systems over p-groups with an abelian subgroup of index p. Results which do not fall into either of the above categories include a new proof of Thompson's normal p-complement Theorem for saturated fusion systems and characterisations of certain quotients of fusion systems which possess a normal subgroup.

No description available.

No description available.