Transitive normality is not quotient-transitive

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This article gives the statement, and possibly proof, of a subgroup property (i.e., transitively normal subgroup) not satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property).
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Statement

Property-theoretic statement

The property of being a transitively normal subgroup is not a quotient-transitive subgroup property.

Statement with symbols

It is possible to have the following situation: H \le K \le G, H is a transitively normal subgroup of a group G, and K/H is a transitively normal subgroup of G/H, and K is not a transitively normal subgroup of G.

Definitions used

Transitively normal subgroup

Further information: Transitively normal subgroup

We say that H is a transitively normal subgroup of G if whenever K is a normal subgroup of H, K is also a normal subgroup of G.

Related facts

Failure of quotient-transitivity for related properties

The same example used here applies to all these related properties:

Proof

Example of the dihedral group

Further information: dihedral group:D8

Let G be the dihedral group, given by:

G = \langle a, x \mid a^4 = x^2 = 1, xax^{-1} = a^{-1} \rangle.

Define subgroups:

H = \langle a^2 \rangle, \qquad K = \langle a^2, x \rangle.

H is a subgroup of order two, hence it is transitively normal in G. K/H is a subgroup of order two in G/H, hence it is transitively normal in G.

However, K is not transitively normal in G, because the subgroup \langle x \rangle of K is normal in K but not in G.