2-subnormality is not transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-subnormal subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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A 2-subnormal subgroup of a 2-subnormal subgroup need not be 2-subnormal.
Failures of transitivity
- Normality is not transitive
- There exist subgroups of arbitrarily large subnormal depth
- Descendant not implies subnormal
- Ascendant not implies subnormal
Existence of transiters
Left-transitively 2-subnormal subgroup
Further information: Left-transitively 2-subnormal subgroup
A subgroup of a group is termed left-transitively 2-subnormal in if whenever is 2-subnormal in some group , so is . Since any characteristic subgroup of a normal subgroup is normal, every characteristic subgroup is left-transitively 2-subnormal.
Right-transitively 2-subnormal subgroup
Further information: Right-transitively 2-subnormal subgroup
A subgroup of a group is termed right-transitively 2-subnormal in if whenever is a 2-subnormal subgroup of , is 2-subnormal in . Any transitively normal subgroup, as well as any base of a wreath product, is right-transitively 2-subnormal.
The proof follows directly from fact (1): if every 2-subnormal subgroup of a 2-subnormal subgroup were 2-subnormal, then every subnormal subgroup would be 2-subnormal, and we would not get subgroups of arbitrarily large subnormal depth.