There exist infinite nilpotent groups in which every automorphism is inner
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Contents
Statement
It is possible to have an infinite nilpotent group in which every automorphism is inner: an infinite nilpotent group that is also a group in which every automorphism is inner. In other words, is infinite and nilpotent and every automorphism of is an inner automorphism, or equivalently, the outer automorphism group of is the trivial group.
Related facts
Opposite facts
Group type | Opposite fact | Proved in ... | Implication in our language |
---|---|---|---|
group of prime power order, hence also finite nilpotent group | Group of prime power order is either trivial or of prime order or has outer automorphism class of same prime order | Nichtabelsche $p$-Gruppen besitzen äussere $p$-Automorphismen by Wolfgang Gaschütz, Journal of Algebra, ISSN 00218693, Volume 4, Page 1 - 2(Year 1966): ^{}^{More info} | This shows that with the exception of the trivial group and cyclic group:Z2, every group of prime power order has a nontrivial outer automorphism. To see this, note that if the group is of prime order , the outer automorphism group has order and is this nontrivial for . In other cases, the outer automorphism group has a subgroup of order , so it is nontrivial. |
finite nilpotent group | Finite nilpotent and every automorphism is inner implies trivial or cyclic of order two | (corollary of previous result) | Follows from preceding result and the fact that automorphism group of finite nilpotent group is direct product of automorphism groups of Sylow subgroups. |
finitely generated nilpotent group | Finitely generated nilpotent and every automorphism is inner implies trivial or cyclic of order two | 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} |
Any finitely generated nilpotent group other than the trivial group and cyclic group:Z2 has a nontrivial outer automorphism group. |
nilpotent p-group | nilpotent p-group and every automorphism is inner implies trivial or cyclic or order two | A nilpotent p-group possesses an outer automorphism by A. E. Zalesskii, Dokl. Akad. Nauk SSSR, Volume 196,Number 4, Page 751 - 754(Year 1971): ^{}^{More info} | Any nilpotent p-group other than the trivial group and cyclic group:Z2 has a nontrivial outer automorphism group. |
periodic nilpotent group | periodic nilpotent and every automorphism is inner implies trivial or cyclic of order two | (follows from preceding result) | Any periodic nilpotent group other than the trivial group and cyclic group:Z has a nontrivial outer automorphism group. |
Facts used
- Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
- Homomorphism from abelian group to torsion-free abelian group is completely determined by images of a maximal linearly independent subset
Proof
The proof here is taken directly from the paper that provides the original proof. The group that we construct, called the Zalesskii group, is a subgroup of unitriangular matrix group:UT(3,Q), hence it is a group of nilpotency class two.
Construction of the group
Further information: Zalesskii group
Step no. | Step detail |
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1 | Partition the set of all prime numbers into three pairwise disjoint subsets where , all three of the subsets being infinite. |
2 | Let be the set |
3 | Let be the set |
4 | Let be the set |
5 | The group we are interested in is the subgroup inside , the group of upper-triangular matrices with 1s on the diagonal and rational entries. |
Proof that it works
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
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1 | For any subgroup of , the only automorphisms of are the multiplication maps by where . | Fact (2) | |||
2 | Denote by the subgroup of that is the group of rational numbers with square-free denominators. Then, for any positive integer , the subgroup has index in . | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | |||
3 | The center of is isomorphic to and comprises matrices of the form , with . | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | |||
4 | The quotient group has only two automorphisms: the identity map and the inverse map. | Fact (2) | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | ||
5 | Every automorphism of induces the identity map on and on | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | |||
6 | Every automorphism of is inner. | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
References
Paper with original proof
- An example of a torsion-free nilpotent group having no outer automorphisms by A. E. Zalesskii, Matematicheskie Zametki, Volume 11,Number 1, Page 21 - 26(January 1972): ^{}^{More info}