Normal not implies right-transitively fixed-depth subnormal

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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., right-transitively fixed-depth subnormal subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about normal subgroup|Get more facts about right-transitively fixed-depth subnormal subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not right-transitively fixed-depth subnormal subgroup|View examples of subgroups satisfying property normal subgroup and right-transitively fixed-depth subnormal subgroup
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., right-transitively fixed-depth subnormal subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about subnormal subgroup|Get more facts about right-transitively fixed-depth subnormal subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property subnormal subgroup but not right-transitively fixed-depth subnormal subgroup|View examples of subgroups satisfying property subnormal subgroup and right-transitively fixed-depth subnormal subgroup

Statement

There exists a group G and a normal subgroup K of G such that K is not right-transitively fixed-depth subnormal in G. In other words, for any k \ge 1, there exists a k-subnormal subgroup H of K such that H is not k-subnormal in G.

Related facts

Proof

Example of the infinite dihedral group

Let G be the infinite dihedral group, i.e., the group given by:

G = \langle a,x \mid x^2 = 1, xax^{-1} = a^{-1} \rangle.

Let K = \langle a^2, x \rangle. K is a subgroup of index two in G, and is normal. For any k \ge 1, consider the subgroup:

H = \langle a^{2^{k+1}}, x \rangle.

Then:

  • The subnormal depth of H in K is k. In particular, H is k-subnormal in K.
  • The subnormal depth of H in G is k + 1. In particular, H is not k-subnormal in G.