Comparison of Union-Preserving and Contact Transformations.

a given group then it is not possible for one of them to be contained in the other. If the operators which are commutative with one of the non-invariant operators of one of the groups under consideration do not always constitute an Abelian group then the prime number of the conjugates for each of its sets of conjugate operators of its non-invariant operators must be the same and hence these groups were considered before in these PROCEEDINGS, 32, 53-56 (1946) under a somewhat more general title, which has much in common with the present article. This article aims to clarify some of the common points as well as to extend the types of the groups under consideration. The two articles together aim to consider fundamental questions of all the groups in which all the non-invariant operators appear in sets of coniugates such that each set involves a prime number of operators. It results that the number of different primes involved in the same group cannot exceed two and that when there are two such primes the larger diminished by unity is divisible by the smaller. The fact that there cannot be more than two such prime numbers in a given group seems to deserve special emphasis.