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.
For an automorphism of the vector space to preserve , it must either preserve both parts of the partition, or interchange them. However, since is unitriangular, it cannot send anything in the positive part to anything in the non-positive part. Therefore, it cannot interchange the parts, and therefore it must preserve both parts. We can then show that this forces the normalizing element to be within .
|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|