Finitely generated nilpotent and every automorphism is inner implies trivial or cyclic of order two
From Groupprops
Statement
Suppose is a finitely generated nilpotent group (i.e., it is both a finitely generated group and a nilpotent group) that is also a group in which every automorphism is inner. Then, is either the trivial group or is isomorphic to cyclic group:Z2.
Another way of putting this is that any nilpotent group in which every automorphism is inner must be either the trivial group, or cyclic group:Z2, or an infinite (and non-finitely generated) nilpotent group..
Related facts
Similar facts
- Finite nilpotent and every automorphism is inner implies trivial or cyclic of order two
- Group of prime power order is either trivial or of prime order or has outer automorphism class of same prime order
Opposite facts
References
- The existence of outer automorphism groups of some groups. I by R. Ree, Proceedings of the American Mathematical Society, Volume 7,Number 6, Page 962 - 964(Year 1956): ^{}^{More info}
- The existence of outer automorphism groups of some groups. II by R. Ree, Proceedings of the American Mathematical Society, Volume 9,Number 1, Page 105 - 109(Year 1958): ^{}^{More info}