Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-12-04T10:16:27.036Z Has data issue: false hasContentIssue false
  • English
  • Français

Some finiteness conditions for automorphism groups

Published online by Cambridge University Press:  18 May 2009

Silvana Franciosi  and
Francesco de Giovanni
Silvana Franciosi
Affiliation:
Dipartimento di Matematica “R. Caccioppoll”, Via Mezzocannone 8, 80134 Napoli, Italy
Francesco de Giovanni
Affiliation:
Dipartimento di Matematica “R. Caccioppoll”, Via Mezzocannone 8, 80134 Napoli, Italy
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 29 , Issue 2 , July 1987 , pp. 259 - 265
Copyright
Copyright © Glasgow Mathematical Journal Trust 1987

References

1. Baer, R., Finite extensions of abelian groups with minimum condition, Trans. Amer. Math. Soc. 79 (1955), 521–540.CrossRefGoogle Scholar
2. Dieudonné, J., Les determinants sur un corps non commutatif, Bull. Soc. Math. France 71 (1943), 27–45.CrossRefGoogle Scholar
3. Franciosi, S. and de Giovanni, F., A note on groups with countable automorphism groups, Arch. Math. (Basel), 47 (1986), 12–16.CrossRefGoogle Scholar
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.Google Scholar
5. MacLane, S., Homology (Springer, 1975).Google Scholar
6. Nagrebeckiĭ, V. T., On the periodic part of a group with a finite number of automorphisms, Dokl. Akad. Nauk SSSR 205 (1972), 519–521 = Soviet Math. Dokl. 13 (1972), 953–956.Google Scholar
7. Robinson, D. J. S., Finiteness conditions and generalized soluble groups (Springer, 1972).CrossRefGoogle Scholar
8. Robinson, D. J. S., On the cohomology of soluble groups with finite rank, J. Pure Appl. Algebra 6 (1975), 155–164.CrossRefGoogle Scholar
9. Robinson, D. J. S., Homology of group extensions with divisible abelian kernel, J. Pure Appl. Algebra 14 (1979), 145–165.CrossRefGoogle Scholar
10. Robinson, D. J. S., Infinite torsion groups as automorphism groups, Quart. J. Math. Oxford Ser. (2) 30 (1979), 351–364.CrossRefGoogle Scholar
11. Rosenberg, A., The structure of the infinite general linear group, Ann. of Math. 68 (1958), 278–294.CrossRefGoogle Scholar
12. Silcock, H. L., Metanilpotent groups satisfying the minimal condition for normal subgroups, Math. Z. 135 (1974), 165–173.CrossRefGoogle Scholar
13. Stammbach, U., Homology in Group Theory (Lecture Notes in Mathematics 359, Springer, 1973).CrossRefGoogle Scholar
14. Zimmerman, J., Countable torsion FC-groups as automorphism groups, Arch. Math. (Basel) 43 (1984), 108–116.CrossRefGoogle Scholar