The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups.

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An automorphism σ of a finite group G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G shall be called Coleman automorphism, named for D. B. Coleman, who's important observation from [2] especially shows that such automorphisms occur naturally in the study of the normalizer N of G in the units U of the integral group ZG. Let Out Col (G) be the image of these automorphisms in Out(G). We prove that Out Col (G) is always an abelian group (based on previous work of E. C. Dade, who showed that Out Col (G) is always nilpotent). We prove that if no composition factor of G has order p (a fixed prime), then Out Col (G) is a p -group. If O p (G) = 1, it suffices to assume that no chief factor of G has order p. If G is solvable and no chief factor of G/O 2 (G) has order 2, then N = GZ, where Z is the center of U. This improves an earlier result of S.

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