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These notes derive from a course of lectures delivered at the University of Florida in Gainesville during 1971/2. Dr Gagen presents a simplified treatment of recent work by H. Bender on the classification of non-soluble groups with abelian Sylow 2-subgroups, together with some background material of wide interest. The book is for research students and specialists in group theory and allied subjects such as finite geometries.

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We say that a subgroup H is an w-th maximal subgroup of G if there exists a chain of subgroups REMARK. If the group PSL(2, q) satisfies the condition that all third maximal subgroups are nilpotent then it follows that (a) q = 2 r , 3* or t, where r, s, t are primes, r > 2, and (b) if q -3 r or 2 s , then {q+l)je, {q-l)/e, where e = 2 if q is odd, and e = 1 if q is even, are products of at most two (not necessarily distinct) primes; if q = t, then (t-1)/2 is a product of at most two primes and (i+l)/2 is either a product of at most two primes or a power of 2.Conversely, the groups Sz(2 3 ) and PSL (2,q), where q satisfies the conditions (a) and (b) above, have the property that every third maximal subgroup is nilpotent.

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