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This paper will be concerned mainly with automorphisms of groups. The concept of a group endomorphism will be used at various points in this paper.

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.

For a finite group G = {g_0 = 1, g_1,. . ., g_{n-1}} , we can associate independent variables x_0, x_1, . . ., x_{n-1} where x_i = x_{g_i}. There is a natural action of Aut(G) on C[x_0, . . . ,x_{n-})]. Let C_1, . . . , C_r be the conjugacy classes of G. If C = {g_{i_1}, g_{i_2}, . . . , g_{i_u }} is a conjugacy class, then let x(C) = x_{i_1} + x_{i_2} + . . . + x_{i_u}. Let ρG be the representation of Aut(G) on C[x_0, . . . , x_(n-1)]/〈x(C_1), . . . , x(C_r) 〉 and let Χ_G be the character afforded by ρ_G. If G is a dihedral group of the form D_2p, D_4p or D_{2p^2}, with p an odd prime, I show how Χ_G splits into irreducible constituents. I also show how the module C[x_0, . . . ,x_{n-1}]/ decomposes into irreducible submodules. This problem is motivated by results of Humphries [2] relating to random walks on groups and the group determinant.

A group G is called locally finite if every finitely generated subgroup of G is finite. In this thesis we study the centralizers of subgroups in simple locally finite groups. Hartley proved that in a linear simple locally finite group, the fixed point of every semisimple automorphism contains infinitely many elements of distinct prime orders. In the first part of this thesis, centralizers of finite abelian subgroups of linear simple locally finite groups are studied and the following result is proved: If G is a linear simple locally finite group and A is a finite d-abelian subgroup consisting of semisimple elements of G, then C_G(A) has an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes pi. Hartley asked the following question: Let G be a non-linear simple locally finite group and F be any subgroup of G. Is CG(F) necessarily infinite? In the second part of this thesis, the following problem is studied: Determine the nonlinear simple locally finite groups G and their finite subgroups F such that C_G(F) contains an infinite abelian subgroup which is isomorphic to the direct product of cyclic groups of order pi for infinitely many distinct primes p_i. We prove the following: Let G be a non-linear simple locally finite group with a split Kegel cover K and F be any finite subgroup consisting of K-semisimple elements of G. Then the centralizer C_G(F) contains an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes p_i.