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Some finiteness conditions for automorphism groups

  • Silvana Franciosi (a1) and Francesco de Giovanni (a1)

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Many authors have investigated the behaviour of the elements of finite order of a group G when finiteness conditions are imposed on the automorphism group Aut G of G. The first result was obtained in 1955 by Baer [1], who proved thata torsion group with finitely many automorphisms is finite. This theorem was generalized by Nagrebeckii in [6], where he proved that if the automorphism group Aut G is finite then the set of elements of finite order of G is a finite subgroup.

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References

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1.Baer, R., Finite extensions of abelian groups with minimum condition, Trans. Amer. Math. Soc. 79 (1955), 521540.
2.Dieudonné, J., Les determinants sur un corps non commutatif, Bull. Soc. Math. France 71 (1943), 2745.
3.Franciosi, S. and de Giovanni, F., A note on groups with countable automorphism groups, Arch. Math. (Basel), 47 (1986), 1216.
4.Franciosi, S., de Giovanni, F. and Robinson, D. J. S., On torsion in groups whose automorphism groups have finite rank, Rocky Mountain J. Math., to appear.
5.MacLane, S., Homology (Springer, 1975).
6.Nagrebeckiĭ, V. T., On the periodic part of a group with a finite number of automorphisms, Dokl. Akad. Nauk SSSR 205 (1972), 519521 = Soviet Math. Dokl. 13 (1972), 953–956.
7.Robinson, D. J. S., Finiteness conditions and generalized soluble groups (Springer, 1972).
8.Robinson, D. J. S., On the cohomology of soluble groups with finite rank, J. Pure Appl. Algebra 6 (1975), 155164.
9.Robinson, D. J. S., Homology of group extensions with divisible abelian kernel, J. Pure Appl. Algebra 14 (1979), 145165.
10.Robinson, D. J. S., Infinite torsion groups as automorphism groups, Quart. J. Math. Oxford Ser. (2) 30 (1979), 351364.
11.Rosenberg, A., The structure of the infinite general linear group, Ann. of Math. 68 (1958), 278294.
12.Silcock, H. L., Metanilpotent groups satisfying the minimal condition for normal subgroups, Math. Z. 135 (1974), 165173.
13.Stammbach, U., Homology in Group Theory (Lecture Notes in Mathematics 359, Springer, 1973).
14.Zimmerman, J., Countable torsion FC-groups as automorphism groups, Arch. Math. (Basel) 43 (1984), 108116.

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