Let A, B, C be respectively the class of all finite supersolvable groups, the class of all finite groups which satisfy the converse to Lagrange's theorem, and the class of all finite solvable groups.We show that Aa B aC, and give examples to show that both of the inclusions are actually proper.Vt will denote pairwise distinct positive integer primes.If G and H are finite groups, then, ζ G" will denote the commutator subgroup of G, 'G x if' will denote the external direct product of G and H, and '\' will denote the order of G. Ά 4 9 will denote the alternating group on 4 symbols, V will denote the identity of A 4 , and 4 C 2 ' will denote the cyclic group of order 2.We are concerned here only with finite groups; throughout, when we say 'group', we intend this to be read as 'finite group', and 'G' will always denote a finite group.Our version of the converse to Lagrange's theorem is as follows:
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Henry Bray (1968) studied this question.