When is aut g abelian

I have more notes on groups whose automorphism group is abelian here and here. There have been some activities on this topic recently. A class of such groups is constructed in "V. Jain, M. Yadav, On finite p-groups whose automorphisms are all central, Israel J. Sign up to join this community. The best answers are voted up and rise to the top. When is Aut G abelian Ask Question.

Asked 11 years, 10 months ago. Active 3 years ago. Viewed 8k times. Improve this question. Martin Sleziak 4, 3 3 gold badges 29 29 silver badges 37 37 bronze badges. Martin Brandenburg Martin Brandenburg Add a comment. Active Oldest Votes. FYI, I found this via a google search. Improve this answer. Pete L. Clark Pete L. Clark For such questions I admit to following the advice of SO immortal Jon Skeet: submit a brief version first and then edit to add the desired frills.

For finite groups, being abelian and the automorphism group being abelian as well implies cyclic. In the infinite case, there are locally cyclic groups that are not cyclic, and these have abelian automorphism groups. For instance, the additive group of rational numbers has an abelian automorphism group the multiplicative group of rational numbers. Vipul Naik Vipul Naik 6, 2 2 gold badges 28 28 silver badges 73 73 bronze badges.

Manoj K. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. The first implication is also true, i. In general however, a group with abelian automorphism group need not be abelian, i.

See the article Jonah, D. Basel 26 , In the infinite case, there are locally cyclic groups that are not cyclic, and these have abelian automorphism groups. For instance, the additive group of rational numbers has an abelian automorphism group the multiplicative group of rational numbers. Sign up to join this community. The best answers are voted up and rise to the top.

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