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• W2067096161 abstract "We prove that if G is a group such that Aut G is a countably infinite torsion FC-group, then Aut G contains an infinite locally soluble, normal subgroup and hence a nontrivial abelian normal subgroup. It follows that a countably infinite subdirect product of nontrivial finite groups, of which only finitely many have nontrivial abelian normal subgroups, is not the automorphism group of any group. We are concerned with the question: What classes of torsion groups can occur as the full group of automorphisms Aut G of a group G? Robinson [1] has shown that if Aut G is a Cernikov group (a finite extension of a radicable abelian group with the minimal condition), then Aut G is finite. He has also shown that if Aut G is a nilpotent torsion group, then Aut G has finite exponent. The case where G is a group such that Aut G is a countable torsion FC-group (finite conjugate) was examined in a previous paper [2]. It was shown that if G is a group such that Aut G is a countable torsion FC-group, then Aut G has finite exponent if either (1) Aut G has min-2 or (2) v(Aut G) is finite, where vT(H) is the set of all primes dividing the order of some torsion element of H. In addition, an example of a countable torsion FC-group of infinite exponent which occurred as an automorphism group was given to show that the theorem could not be improved. This example contains a nontrivial abelian normal subgroup. The question arises: Can we find an example which has no nontrivial abelian normal subgroups? We will answer this question in the negative. THEOREM. Let G be a group such that Aut G is a countably infinite periodic FC-group. Then either (a) Aut G contains an infinite abelian normal subgroup N, or (b) Aut G contains an infinite, locally soluble, normal {2, 3}-subgroup of bounded exponent and finite index. In either case, Aut G contains a nontrivial abelian normal subgroup. PROOF. Let Q = G/C -InnG, where C is the center of G, and let T be the torsion subgroup of C. It was proven in [2] that Q and Tp are finite for all primes p. Let q = I and let p be any prime which does not divide 2q. Since Tp is finite, we have C = C1 x Tp. It is well known that since IQI and ITp7 are relatively prime, G splits over Tp. It follows that there exists a group G1 containing C1 such that Received by the editors November 26, 1984 and, in revised form, January 20, 1985. 1980 Mathematics Subject Classification. Primary 20F28, 20E26." @default.
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• W2067096161 date "1986-01-01" @default.
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• W2067096161 title "Some properties of FC-groups which occur as automorphism groups" @default.
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• W2067096161 doi "https://doi.org/10.1090/s0002-9939-1986-0813805-2" @default.
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