Normal versus permutable

Introduction
This article is about the relation, the similarity and contrast, between the well-known subgroup property of normality and the somewhat more obscure subgroup property called permutability: permuting with every subgroup.

Normal subgroup
A subgroup $$H$$ of a group $$G$$ is termed normal if for every $$g \in G$$:

$$Hg = gH$$

In other words, its left cosets equal its right cosets.

Permutable subgroup
A subgroup $$H$$ of a group $$G$$ is termed permutable or quasinormal if for every subgroup $$K$$ of $$G$$:

$$HK = KH$$

In other words, $$H$$ and $$K$$ are permuting subgroups.

It actually suffices to check the condition only for all cyclic subgroups $$K$$ of $$G$$.

Every normal subgroup is permutable
Normality requires the subgroup to commute with every element. Hence, it commutes with every subset, and in particular, with every subgroup.

Normality satisfies intermediate subgroup condition, transfer condition, inverse image condition
It is clear from the definition that if $$H$$ is normal in $$G$$, then $$H$$ is also normal in any intermediate subgroup. It is further clear that, for any subgroup $$K \le G$$, $$H \cap K$$ is normal in $$K$$. Finally, the inverse image of a normal subgroup under any homomorphism is also normal.

Permutability satisfies intermediate subgroup condition, transfer condition, inverse image condition
If $$H$$ is permutable in $$G$$, then $$H$$ is also permutable in any intermediate subgroup. Further, it is true that for any $$K \le G$$, $$H \cap K$$ is permutable in $$K$$. Finally, the inverse image of a permutable subgroup under any homomorphism is permutable.

Neither property is transitive
A normal subgroup of a normal subgroup need not be normal. Similarly, a permutable subgroup of a permutable subgroup need not be permutable.

Left transiter for normality
If $$H$$ is a characteristic subgroup of $$K$$ and $$K$$ is a normal subgroup of $$G$$, then $$H$$ is normal in $$G$$ as well.

In fact, if $$H \le K$$ is a subgroup such that whenever $$K$$ is normal in some bigger group $$G$$, so is $$H$$, we must have that $$H$$ is characteristic in $$K$$.

Left transiter for permutability
It is unclear what the left transiter of permutability should be. In other words, there is no known characterization yet of subgroups $$H$$ of $$K$$ such that whenever $$K$$ is a permutable subgroup of some bigger group $$G$$ containing $$K$$, so is $$H$$.

It is true that if $$H = K$$ or $$H$$ is trivial, this is satisfied. Further, if $$H$$ is left-transitively permutable in $$K$$ (i.e., it satisfies the above property) then $$H$$ must be a characteristic subgroup. However, characteristicity is probably not a sufficient condition.

Permutability is closed under joins but not under intersections
An arbitrary join of permutable subgroups is permutable. However, an intersection of two permutable subgroups need not be permutable. In fact, even if one of the subgroups is normal, the intersection need not be permutable.

Upper join-closedness
If $$H$$ is normal in a collection of intermediate subgroups $$K_i$$ of a group $$G$$, then $$H$$ is also normal in the join of the $$K_i$$s.

The corresponding statement is not true for permutability. If $$H$$ is permutable in intermediate subgroups $$K_1$$ and $$K_2$$ of $$G$$, $$H$$ need not be permutable in the join $$\langle K_1, K_2 \rangle$$.

Corresponding notions of simplicity

 * Simple group is a group that has no proper nontrivial normal subgroups.
 * The analogous notion for permutable subgroups would be a group that has no proper nontrivial permutable subgroups.

It turns out that for a slender group (i.e., a group satisfying an ascending chain condition on all subgroups), the two notions are equivalent. This follows from the fact that any maximal conjugate-permutable subgroup is normal. Since conjugate-permutability is a weaker condition than permutability, the existence of a proper nontrivial permutable subgroup implies the existence of a proper nontrivial conjugate-permutable subgroup, and the ascendin chain condition forces the existence of a maximal proper conjugate-permutable subgroup containing it. This is a proper nontrivial normal subgroup by the given fact, so the existence of a proper nontrivial permutable subgroup implies the existence of a proper nontrivial normal subgroup.

Hereditarily operator and Hamiltonian operator

 * A hereditarily normal subgroup is a normal subgroup such that any subgroup contained in it is normal. It turns out that the center, or more generally, any central subgroup, is a hereditarily normal subgroup. A cyclic normal subgroup is also hereditarily normal.
 * A hereditarily permutable subgroup is a permutable subgroup such that any subgroup contained in it is permutable. It turns out that the Baer norm (defined as the intersection of normalizers of all subgroups), and more generally, any subgroup of the Baer norm, is hereditarily permutable.
 * A Dedekind group is a group in which every subgroup is normal. A Hamiltonian group is a non-abelian Dedekind group. It turns out that the center and the Baer norm are both Dedekind groups (although the Baer norm is Dedekind, every subgroup of it need not be normal in the whole group). All Hamiltonian groups have been classified.
 * A PH-group is a non-abelian group where every subgroup is permutable.

Permutable as sandwiched between normal and subnormal
In a finite group, or more generally in a group of finite composition length, any permutable subgroup is subnormal. In fact, there are many properties sandwiched between permutability and subnormality for finite groups. Among these is the property of being a conjugate-permutable subgroup: a subgroup that permutes with each of its conjugates. This is stronger than subnormality but weaker than permutability. Slightly stronger than conjugate-permutability is the property of being an automorph-permutable subgroup.

Situations where the two notions coincide

 * Group in which every permutable subgroup is normal: This is a group where every permutable subgroup is normal. For finite groups, a T-group, which is a group in which subnormal subgroup is normal, is also a group in which every permutable subgroup is normal.
 * In a finite group, any permutable subgroup that is a subnormal-to-normal subgroup, or an intermediately subnormal-to-normal subgroup, is normal. For instance, any permutable Sylow subgroup is normal, and any permutable Hall subgroup is normal. Any permutable pronormal subroup is normal too.