THE FINITE INNER AUTOMORPHISM-GROUPS OF DIVISION RINGS

Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x→u-1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2.3-2.7, 2.10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2.26), and provide examples of finite groups having faithful irreducible projective representations over fields. Let p be zero or a prime, and let us write Jp for the class of all finite groups G such that there exists a division ring D of characteristic p with centre k such that G embeds into D*/k*. Recall that for a finite group G, a stem cover (H, (ϕ) of G consists of a group H and an epimorphism ϕ: H→G such that ker ϕ ⊆ ζ1(H) ∩ H’, and ker ϕ≃H2(G, C*), the Schur multiplier of G. (As usual, ζ1(H) denotes the centre and H’ the derived group of H.) Of course, H is called a covering group of G. Finally, if p is a prime not dividing the order of a finite cyclic group C, and if Ω is a group of automorphisms of C, each of which acts by raising the elements of C to some power of p, then we refer to Ω as a group of p-Frobenius automorphisms of C. Of course Ω is cyclic since its elements are various powers of the Frobenius automorphism x → xp of C. © 1995, Cambridge Philosophical Society. All rights reserved.