Locally nilpotent not implies normalizer condition
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., locally nilpotent group) need not satisfy the second group property (i.e., group satisfying normalizer condition)
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It is possible to have a group that is a locally nilpotent group (every finitely generated subgroup of it is a nilpotent group) but is not a group satisfying normalizer condition. In other words, has a proper subgroup that is self-normalizing: .
Description of the example
Take a field . Choose as follows:
- If we want a torsion-free example, take , or any other field of characteristic zero.
- If we want an example that is a p-group for some prime number , take to be the field of elements, or alternatively, any field of characteristic .
Let be McLain's group .
Proof that it is locally nilpotent
is locally nilpotent because any finitely generated subgroup of has finite support in the basis , i.e., it moves only finitely many basis elements, so it is in a finite-degree unitriangular matrix group over , which is nilpotent.
Proof that it does not satisfy the normalizer condition
Let be the subgroup of generated by the automorphisms:
Basically, we have partitioned the set into two totally ordered subsets: the non-positive and the positive numbers, and are only considering the automorphisms in that keep each part within itself.
We now need to show that the only automorphisms of the vector space that are in and preserve must be within itself. The basic idea behind the proof (and the key way it differs from the finite case) is that the indexing set is unbounded. In the finite case, the map:
would be forced to commute with everything. But in the infinite case, there are no min and max, so any time something from the non-positive side maps to something that has a component on the positive side, we can find things below the non-positive number and above the positive number to provide a contradiction.PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
|Book||Page number||Chapter and section||Contextual information||View|
|A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info||369||Section 12.2||proof in exercise 12.2.8||Google Books|