Normal versus permutable

This survey article compares, and contrasts, the following subgroup properties: normal subgroup versus permutable subgroup
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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.

Definitions

Normal subgroup

Further information: Normal subgroup

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

$H g = g H$

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

Permutable subgroup

Further information: Permutable subgroup

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

$H K = K H$

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

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

Implication relations

Every normal subgroup is permutable

Further information: Normal implies permutable

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

Every permutable subgroup need not be normal

Further information: Permutable not implies normal

Similar behavior with respect to intermediate subgroups

Normality satisfies intermediate subgroup condition, transfer condition, inverse image condition

Further information: Normality satisfies intermediate subgroup condition, Normality satisfies transfer condition, Normality satisfies 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 $K \leq G$ , $H \cap K$ is normal in $K$ . Finally, the inverse image of a normal subgroup under any homomorphism is also normal.

Permutability satisfies intermediate subgroup condition, trasnfer condition, inverse image condition

Further information: Permutability satisfies intermediate subgroup condition, Permutability satisfies transfer condition, Normality satisfies 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 $K \leq G$ , $H \cap K$ is permutable in $K$ . Finally, the inverse image of a permutable subgroup under any homomorphism is permutable.

Transitivity and transiters

Neither property is transitive

Further information: Normality is not transitive, Permutability is not transitive

A normal subgroup of a normal subgroup need not be normal. Similarly, a permutable subgroup of a permutable subgroup need not be permutable.

Normality has left and right transiters

The lack of transitivity of normality can be remedied somewhat. In particular:

Any characteristic subgroup of a normal subgroup is normal: Thus, for instance, the center and the commutator subgroup of a normal subgroup are normal.
Any normal subgroup of a so-called transitively normal subgroup, is normal. In particular, any central subgroup, hereditarily normal subgroup, direct factor or central factor is transitively normal.

Permutability has no clearly identifiable left or right transiters

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Effect of joins and intersections

Normality is closed under both

Further information: Normality is strongly join-closed, normality is strongly intersection-closed

Permutability is closed under joins but not under intersections

Further information: Permutability is strongly join-closed, Permutability is not intersection-closed

Upper join-closedness

Corresponding notions of simplicity

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