| Summary: | <p>In this thesis, a study is made of finite groups which satisfy the following hypothesis:-<br/> <p>(*) G is a finite group admitting an automorphism α of order r with a fixed point subgroup of order q, where q and r are distinct prime numbers.</p> <p>The main result is</p> <p><strong>Theorem A</strong>. Let G be a group satisfying hypothesis (*). Assume either that q is odd or that q = 2 and r is not a Fermat prime greater than 3. Then G is soluble.</p> <p>First a structure theorem for soluble groups satisfying hypothesis (*) is obtained (theorem B). Solubility of groups satisfying hypothesis (*) is shown under the additional assumption that the symmetric group S<sub>4</sub> is not involved: this serves to prove solubility in general for r = 2 or r =3, and to point to the initial reductions in the general case. The proof of theorem A is by induction on the order of groups satisfying the hypothesis for a given pair (q,r). A minimal counterexample is simple, and the remainder of the proof is to show non-existence. After initial reductions the cases q odd and q = 2 are considered separately.</p> <p>For the case q odd, we require the following theorem which is of independent interest: it is a generalisation of a result of Glauberman, theorem A of (<strong>12</strong>).</p> <p>Let p be a prime and P a p-group. Let d(P) denote the largest of the orders of the abelian subgroups of P, and let J(P) be the subgroup of P generated by the abelian subgroups of order d(P). Also, let Qd(p) be the natural semi-direct product of z<sub>p</sub> × z<sub>p</sub>, regarded as a vector space, by SL(2,p). We obtain</p> <p><strong>Theorem 9.3</strong>. Let G be a finite group, p a prime, P a Sylow p-subgroup of G, and Q a subgroup of Z(P). If Q is normal in N<sub>G</sub>(J(P)) and if either (i) p is odd and (p-1) does not divide the index [N(Q):C(Q)] or (ii) Qd(p) is not involved in G, then Q is weakly closed in P with respect to G.</p> <p>For the case q = 2, the arguments involved in the proof of theorem A are mainly character-theoretic: we also obtain information about the exceptional cases.</p> <p>Except as mentioned above, the arguments are entirely group-theoretic.</p></p>
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