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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., left-transitively fixed-depth subnormal subgroup)
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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subnormal subgroup) need not satisfy the second subgroup property (i.e., left-transitively fixed-depth subnormal subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications |Get help on looking up subgroup property implications/non-implications
Get more facts about subnormal subgroup| Get more facts about left-transitively fixed-depth subnormal subgroup

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Contents 1 Statement 2 Related facts 3 Proof 3.1 Example of the dihedral group

Statement

It is possible to have a group K and a normal subgroup H such that for every , there exists a group G containing K as a k-subnormal subgroup, but in which H is not a k-subnormal subgroup.

Note that this also gives an example where K is a subnormal subgroup of H, and for every , there exists a group G containing K as a k-subnormal subgroup, but in which H is not a k-subnormal subgroup.

Related facts

• Left transiter of normal is characteristic
• Cofactorial automorphism-invariant implies left-transitively 2-subnormal
• Normality is not transitive
• There exist subgroups of arbitrarily large subnormal depth
• Ascendant not implies subnormal, descendant not implies subnormal
• Normal not implies right-transitively fixed-depth subnormal

Proof

Example of the dihedral group

Further information: dihedral group:D8

Let K be the dihedral group of order eight, given by:

.

Let H be the subgroup of K generated by a² and x. H has index two in K and is hence normal. (subgroup of index two is normal).

For any define G as the group:

.

In other words, G is a dihedral group of order 2^{k + 3}. Consider K as a subgroup of G by identifying and x = x. Then:

• The subnormal depth of K in G is k.
• The subnormal depth of H in G is k + 1.

This completes the proof.