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Composition factors from the group ring and Artin’s theorem on orders of simple groups. (English) Zbl 0668.20009

The paper begins by showing that the integral group ring of a finite group determines the isomorphism types of the factors of each chief series of the group. The normal subgroup correspondence shows that the chief factors and their orders may be recognized from the integral group ring. The isomorphism type is then obtained by application of the generalization of Artin’s theorem on the orders of finite simple groups to characteristically simple groups. An alternative argument is presented; it is based on the fact that each finite simple group has a cyclic Sylow subgroup, a fact whose proof is indicated. Corollaries relevant to the isomorphism problem for group rings are drawn such as that alluded to in the title.
The rest of the paper supplies proofs of the completion of Artin’s theorem to the full list of finite simple groups and of its generalization. The proofs depend on the classification of finite simple groups. The theorem itself states that a finite simple group is determined by its order except for two cases: \(A_ 8\approx L_ 4(2)\) and \(L_ 3(4)\) of order 20160; \(O_{2n+1}(q)\) and \(S_{2n}(q)\) for \(n\geq 3\) and q an odd prime power. Its generalization states that a direct power of finite simple group is determined by its order with exceptions from the same sources. The completion of Artin’s theorem was effected by J. Tits while the generalization is due to P. Cameron and D. N. Teague.
The proofs given here adapt and make explicit certain functions of a natural number variable which Artin used implicitly. These functions are sufficiently discriminating that, constructively from their values, the argument can be retrieved, an order recognized as that of a simple group and the simple group identified (up to the limits in Artin’s theorem). For a natural number N, not a prime power, these functions are: the prime p for which the p-part of N is maximal; the exponent \(\ell\) of p as a divisor of N; the largest order \(\omega\) of p modulo another prime divisor of N and the next largest such order \(\psi\) (if available); the quotient \(\lambda\) of log \(p^{\ell}\) by log N.
The proofs rely on eleven tables which indicate the values of these functions for the orders of finite simple groups and on three ancillary tables concerned with cyclotomic factorization of the order of a group of Lie type. There is one further table which gives all known orders of simple groups not divisible by a prime to the first power only (mainly the squares which are orders of symplectic groups as determined by Newman, Shanks and Williams). Much of the detail in the proofs is elementary number theory but various active conjectures in prime and transcendental number theory are encountered incidentally.
Reviewer: R.Sandling

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
16S34 Group rings
20D08 Simple groups: sporadic groups
20G40 Linear algebraic groups over finite fields
11A07 Congruences; primitive roots; residue systems
11A41 Primes
11D41 Higher degree equations; Fermat’s equation
11A25 Arithmetic functions; related numbers; inversion formulas
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