Some problems in the theory of finite insoluble groups - Collins, M (1969)
In this thesis, a study is made of finite groups which satisfy the following hypothesis:- (*) 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. The main result is Theorem A. 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. 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 S4 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. 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 (12). 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 zp × zp, regarded as a vector space, by SL(2,p). We obtain Theorem 9.3. 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 NG(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. 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. Except as mentioned above, the arguments are entirely group-theoretic.