ON CANONICAL CONSTRUCTIONS. Ill

In the first paper in this series a structure S was called canonically reconstructible from a structure T provided 5 was canonically isomorphic to a structure T' derived from T. The principle proposition of the paper was the scholium which asserted: If a structure 5 contains a substructure T and if S is canonically reconstructible from T, then every automorphism of T may be extended uniquely to an inner automorphism of 5 (relative to the derivation of 5 from T). To illustrate the concepts it was proved in that paper that every automorphism of the group of permutations S (A7) of a set N (finite or infinite) was inner in the group theoretic sense except when N has 6 elements. Let A{N) now denote the subgroup of 2(Ar) generated by all 3-cycles, i.e., all permutations moving exactly 3 elements of N. This will be called the alternating group on N. It will here be shown that the set N can be canonically reconstructed from any subgroup G of S (A7) which contains A{N) when N has 4, 5, or more than 6 elements. It follows by the principles of On canonical constructions, I, that every automorphism of G can then be extended uniquely to an automorphism, i.e., permutation, of N, which in turn induces an inner automorphism of 2{N). This is trivially true when N has 1 or 2 elements. Therefore, it is the case, except when N has cardinality 3 or 6, that every automorphism of G can be extended uniquely to an inner automorphism of S (A7), which is the principle theorem of this paper.1