Normality is not transitive

Statement
There can be a situation where $$H$$ is a normal subgroup of $$K$$ and $$K$$ is a normal subgroup of $$G$$ but $$H$$ is not a normal subgroup of $$G$$.

Transitivity-forcing operator

 * A group in which every normal subgroup of a normal subgroup is normal is termed a T-group. Note that abelian groups and Dedekind groups are T-groups, and any nilpotent group that is a T-group is also a Dedekind group.
 * A group $$K$$ has the property that whenever $$K$$ is normal in $$G$$, every normal subgroup of $$K$$ is normal in $$G$$ (in other words, transitivity holds with $$K$$ as the middle group) if and only if $$K$$ is a group in which every normal subgroup is characteristic.
 * There is no nontrivial group $$G$$ such that whenever $$G$$ is a normal subgroup of a normal subgroup of some group, $$G$$ is normal in that group. In fact, the general example we construct here shows precisely that.

Left transiter
While normality is not transitive, it is still true that every characteristic subgroup of a normal subgroup is normal. Characteristicity is the left transiter of normality -- it is the weakest property $$p$$ such that every subgroup with property $$p$$ in a normal subgroup is normal.

Right transiter
While normality is not transitive, every normal subgroup of a transitively normal subgroup is normal. Being transitively normal is the right transiter of being normal. Properties like being a direct factor, being a central subgroup, and being a central factor imply being transitively normal.

Subnormality
The lack of transitivity of normality can also be remedied by defining the notion of subnormal subgroup. Subnormality is the weakest transitive subgroup property implied by normality. More explicitly, a subgroup $$H$$ is subnormal in a group $$G$$, if we can find a chain of subgroups going up from $$H$$ to $$G$$, with each subgroup normal in its successor.

A special case of this is the notion of 2-subnormal subgroup, which is a normal subgroup of a normal subgroup. Another special case is the notion of a 3-subnormal subgroup, which is a normal subgroup of a normal subgroup of a normal subgroup.

There are also related notions of hypernormalized subgroup, 2-hypernormalized subgroup, ascendant subgroup, descendant subgroup, and serial subgroup.

Corollaries

 * Normality is not a finite-relative-intersection-closed subgroup property, because finite-relative-intersection-closed implies transitive.

Making normality transitive
For simplicity, we assume $$H \le K \le G$$, with $$H$$ the bottom group, $$K$$ the middle group, and $$G$$ the top group.

For particular kinds of groups
For simplicity, we refer below to the three groups as $$H \le K \le G$$, with $$H$$ the bottom group, $$K$$ the middle group, and $$G$$ the top group, such that $$H$$ is normal in $$K$$ and $$K$$ is normal in $$G$$, but $$H$$ is not normal in $$G$$.

General tricks

 * Disproving transitivity
 * Using dihedral groups as counterexamples

Proof
(Also see ).

Generic example
A natural example is as follows. Take any nontrivial group $$H$$, and consider the square, $$K = H \times H$$ (the uses as intermediate construct::external direct product of $$H$$ with itself). Now, consider the uses as intermediate construct::external semidirect product of this group with the group $$\mathbb{Z}/2\mathbb{Z}$$ (the cyclic group of two elements) acting via the exchange automorphism (the automorphism that exchanges the coordinates). Call the big group $$G$$.

(Note that $$G$$ can be described more compactly as the uses as intermediate construct::external wreath product of $$H$$ with the group of order two acting regularly.)

Let $$H_1, H_2$$ be the copies of $$H$$ embedded in $$K$$ as $$H \times \{ e \}$$ and $$\{ e \} \times H$$. We then have:


 * $$H_1$$ is normal in $$K$$: In fact, $$H_1$$ is a direct factor of $$K$$, and is thus normal.
 * $$K$$ is normal in $$G$$: $$K$$ is the base of a semidirect product, and is thus normal. Equivalently, any inner automorphism of $$G$$ is the composite of an inner automorphism in $$K$$ and the exchange automorphism, both of which preserve $$K$$.
 * $$H_1$$ is not normal in $$G$$: The exchange automorphism is an inner automorphism of $$G$$, and it exchanges $$H_1$$ and $$H_2$$ -- in particular, it does not preserve $$H_1$$. Thus, $$H_1$$ is not normal in $$G$$.

Note that both $$H_1$$ and $$H_2$$ are copies of $$H$$, and hence either can be viewed as the fact about::base of a wreath product in $$G$$.

Specific realizations of this generic example
The smallest case of this yields $$H_1$$ a group of order two, and $$G$$ a group of order eight. In fact, here $$G$$ is the dihedral group of order eight and $$H$$ is a cyclic group of order two, with $$H_1$$ and $$H_2$$ being subgroups of order two generated by reflections. Here's how this relates to the usual definition of the dihedral group:

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

$$H_1 = \langle x \rangle, \qquad H_2 = \langle a^2 x \rangle, \qquad K = \langle x, a^2 \rangle$$.

Here, $$H_1$$ and $$H_2$$ are both normal in $$K$$, which is normal in $$G$$, but neither $$H_1$$ nor $$H_2$$ is normal in $$G$$.

For more on the subgroup structure of the dihedral group, refer subgroup structure of dihedral group:D8, particular example::Klein four-subgroups of dihedral group:D8, and particular example::non-normal subgroups of dihedral group:D8.

Implementation of the generic example
Here is an implementation of the generic example, with any nontrivial group $$H$$. Note that you need to define $$H$$ for GAP before executing the commands in this example! Double semicolons have been used to suppress the output here, since the output depends on the choice of $$H$$ (you can use single semicolons instead to display all the outputs).

We first construct the groups $$H_1, H_2, K, G$$ using the wreath product command:

gap> G := WreathProduct(H,SymmetricGroup(2));; gap> H1 := Image(Embedding(G,1));; gap> H2 := Image(Embedding(G,2));; gap> K := Group(Union(H1,H2));;

Next, we check that $$H_1$$ and $$H_2$$ are subgroups of $$K$$ and $$K$$ is a subgroup of $$G$$:

gap> IsSubgroup(G,K); true gap> IsSubgroup(K,H1); true gap> IsSubgroup(K,H2); true

Finally, we check that $$H_1, H_2$$ are both normal in $$K$$ and $$K$$ is normal in $$G$$, but $$H_1$$ and $$H_2$$ are not normal in $$G$$.

gap> IsNormal(G,K); true gap> IsNormal(K,H1); true gap> IsNormal(K,H2); true gap> IsNormal(G,H1); false gap> IsNormal(G,H2); false

The implementation in some special cases
Here is the implementation when $$H$$ is cyclic of order two:

gap> G := WreathProduct(H,SymmetricGroup(2));  gap> H1 := Image(Embedding(G,1)); gap> H2 := Image(Embedding(G,2)); gap> K := Group(Union(H1,H2)); gap> IsSubgroup(G,K); true gap> IsSubgroup(K,H1); true gap> IsSubgroup(K,H2); true gap> IsNormal(G,K); true gap> IsNormal(K,H1); true gap> IsNormal(K,H2); true gap> IsNormal(G,H1); false gap> IsNormal(G,H2); false

Textbook references

 * Also, Page 6 (first mention), and Page 17 (further explanation)
 * Also, Page 135, with justification of the related fact that characteristic of normal implies normal
 * , Page 66
 * Also: Page 28, Page 63