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Abelian Groups pp 655-671 | Cite as

Automorphism Groups

  • László Fuchs
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

Needless to say, automorphism groups contain in general much less information about the group structure than the endomorphism rings. There is an enormous contrast between torsion and torsion-free groups, perhaps even larger than from the point of view of endomorphism rings. Here again, the case of p-groups is more favorable (at least for p > 2) inasmuch as their automorphism groups determine the groups up to isomorphism. The torsion-free case seems uncontrollable, but a close examination shows that, interestingly, only a handful of finite groups may occur as automorphism groups of torsion-free groups of finite rank.

We remark at the outset that we will not give complete proofs of the two most important theorems on automorphism groups: that \(\mathop{\mathrm{Aut}}\nolimits A\) determines A if A is a p-group (p > 2), and the full description of finite automorphism groups of finite rank torsion-free groups. The proofs are too long and require lot of arguments on non-commutative groups.

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • László Fuchs
    • 1
  1. 1.Mathematics DepartmentTulane UniversityNew OrleansUSA

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