Normalizer of 2-subnormal subgroup may have arbitrarily large subnormal depth
This is a statement of the form: there exist subnormal subgroups of arbitrarily large subnormal depth satisfying certain conditions.
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- Abnormal normalizer and 2-subnormal not implies normal: We can have a 2-subnormal subgroup that is not normal, and whose normalizer is abnormal -- very much the opposite of being subnormal.
- There exist subgroups of arbitrarily large subnormal depth
- Subnormality satisfies image condition: The image of a -subnormal subgroup under a surjective homomorphism is -subnormal in the image.
- Subnormality satisfies inverse image condition: The inverse image of a -subnormal subgroup for any homomorphism is -subnormal.
The construction is as follows:
- (uses: Fact (1)): Find a finite group and a subgroup of such that is -subnormal in but is not -subnormal in .
- Let be the additive group of the group ring of over any finite field, and consider the semidirect product where acts by left multiplication. Then, is an Abelian normal subgroup of , so any subgroup of is 2-subnormal in .
- Let be the subspace of spanned by the basis elements corresponding to . is 2-subnormal in , and its normalizer in is . Thus, we have a 2-subnormal subgroup whose normalizer in the whole group is the subgroup .
- Consider the projection map from to with kernel .
- By fact (2), we observe that if is -subnormal in , then is -subnormal in . Thus, is not -subnormal in .
- By fact (3), we find that since is -subnormal in , its full inverse image is -subnormal in .