Pronormal subgroup

Equivalent definitions in tabular format
Under the right action convention, the right action of $$g$$ on $$H$$ by conjugation gives the subgroup $$H^g = g^{-1}Hg$$. Under the left action convention, the left action of $$g$$ on $$H$$ by conjugation gives the subgroup $${}^gH = gHg^{-1}$$. Note that although the actions differ, the notion of being conjugate subgroups inside an intermediate subgroup remains unchanged. This can be explained by the fact that the inverse map reverses the roles of left and right while preserving subgroups.

Equivalence of definitions
The two definitions are equivalent because being conjugate inside the smallest possible intermediate subgroup, i.e., $$\langle H,H^g \rangle$$, implies being conjugate in any intermediate subgroup.

Origin
The notion of pronormal subgroup was introduced by Philip Hall and the first nontrivial results on it were obtained by John S. Rose in his paper Finite soluble groups with pronormal system normalizers.

Extreme examples

 * 1) Every group is pronormal as a subgroup of itself
 * 2) The trivial subgroup is always pronormal.

Generic examples

 * 1) All Sylow subgroups are pronormal.
 * 2) Maximal subgroups and normal subgroups are pronormal.

Particular examples

 * 1) High occurence example: In the symmetric group of degree three, all subgroups are pronormal.
 * 2) Low occurrence example: In a nilpotent group, the pronormal subgroups are the same as the normal subgroups. That's because every subgroup is subnormal, and pronormal and subnormal implies normal.

Non-examples

 * 1) In a symmetric group of order four, or in a symmetric group of higher order, a subgroup generated by a transposition is not pronormal. That's because conjugating it can give a subgroup generated by a disjoint transposition.
 * 2) A subnormal subgroup that is not normal, cannot be pronormal. That's because pronormal and subnormal implies normal.

Formalisms
Pronormality can be expressed using a monadic second-order sentence. The sentence is somewhat complicated. First, note that, using monadic second-order logic, it is possible to construct the subgroup generated by any subset (namely as the smallest subset containing that subset and closed under group operations). Thus, if $$H$$ is a subgroup of $$G$$, the group $$\langle H,gHg^{-1} \rangle$$ can be constructed using monadic second-order logic. Pronormality testing is now the following sentence:

$$\forall g \in G \ \exists x \in \langle H,gHg^{-1} \rangle : \ (\forall y \in H \exists z \in H . gyg^{-1} = xzx^{-1}) \land (\forall v \in H \exists w \in H . gwg^{-1} = xvx^{-1})$$

Related survey articles

 * Subnormal-to-normal and normal-to-characteristic: This survey article discusses in detail a number of subgroup properties that, along with subnormality, imply normality. Pronormality is a prominent one among them. The article also has this implication diagram that shows the implication relations between various subgroup properties.

Weaker properties
For a survey article exploring these properties in greater depth, refer: subnormal-to-normal and normal-to-characteristic



Conjunction with other properties

 * Any pronormal subnormal subgroup is normal, and conversely, a normal subgroup is both pronormal and subnormal:
 * A subgroup is intermediately isomorph-conjugate if and only if it is both pronormal and isomorph-conjugate.

Related group properties

 * Group in which every subgroup is pronormal: Such groups are, in particular, T*-groups, and T-groups: normality is transitive on subgroups of such groups.
 * Group in which every weakly pronormal subgroup is pronormal: All solvable groups, and more generally, all hyper-N-groups, satisfy this condition.
 * Group in which every pronormal subgroup is normal: This property is satisfied by all locally nilpotent groups. For finite groups, it is equivalent to being a finite nilpotent group.

Metaproperties
Pronormality satisfies the intermediate subgroup condition, that is, any pronormal subgroup is pronormal in every intermediate subgroup.

If $$H \le G$$ is pronormal and $$K \le G$$, then $$H \cap K$$ need not be pronormal in $$K$$.

An intersection of pronormal subgroups need not be pronormal. In fact, even a finite intersection of pronormal subgroups need not be pronormal.

The property of pronormality is not transitive. This follows because every normal subgroup is pronormal and every pronormal subnormal subgroup is normal. The proof generalizes to all properties sandwiched between normality and the property of being subnormal-to-normal.

The subordination of this property is the property of being subpronormal.

The normalizer of a pronormal subgroup of a group is pronormal. In fact, it is an abnormal subgroup -- a stronger condition.

If $$H, K$$ are pronormal subgroups of a group $$G$$ such that $$K \le N_G(H)$$, then the join $$HK$$ is also a pronormal subgroup.

A join of pronormal subgroups need not be pronormal. In fact, a join of finitely many pronormal subgroups need not be pronormal.

If $$H$$ is a subgroup of $$G$$ and $$K, L$$ are intermediate subgroups containing $$H$$ such that $$H$$ is pronormal in both $$K$$ and $$L$$, it is not necessary that $$H$$ is pronormal in $$\langle K, L \rangle$$.

If, for a subgroup $$H$$ of a group $$G$$, there exists a unique largest subgroup $$K$$ in which $$H$$ is pronormal, $$K$$ is termed a pronormalizer for $$H$$, and $$H$$ is termed a subgroup having a pronormalizer.

Suppose $$\varphi:G \to K$$ is a surjective homomorphism of groups. Then, if $$H$$ is a pronormal subgroup of $$G$$, $$\varphi(H)$$ is a pronormal subgroup of $$K$$.

If $$H \le K \le G$$ such that $$H$$ is a normal (hence also pronormal) subgroup of $$G$$ and $$K/H$$ is pronormal in $$G/H$$, then $$K$$ is a pronormal subgroup of $$G$$.

Effect of property operators
A subgroup $$H$$ of a group $$G$$ is termed a subpronormal subgroup of $$G$$ if there exists a sequence $$H = H_0 \le H_1 \le H_2 \le \dots \le H_n = G$$ with each $$H_{i-1}$$ a pronormal subgroup in $$H_i$$. Any subgroup of a finite group is subpronormal.

A subgroup $$H$$ of a group $$G$$ is termed right-transitively pronormal in $$G$$ if any pronormal subgroup of $$H$$ is pronormal in $$G$$. Any SCAB-subgroup is right-transitively pronormal.

A subgroup $$H$$ of a group $$G$$ is termed join-transitively pronormal in $$G$$ if the join of $$H$$ with any pronormal subgroup of $$G$$ is pronormal.

A subgroup $$H$$ of a group $$G$$ is termed hereditarily pronormal if every subgroup of $$H$$ is pronormal in $$G$$. Note that this is equivalent to being a right-transitively pronormal subgroup that is also a group in which every subgroup is pronormal.

A subgroup $$H$$ of a group $$G$$ is termed a join of pronormal subgroups in $$G$$ if there is a set of pronormal subgroups of $$G$$ whose join in $$H$$.

Testing
While there is no built-in GAP command for testing pronormality, the test can be accomplished by a short piece of GAP code, available at GAP:IsPronormal. The code is invoked as follows:

IsPronormal(group,subgroup);