Abnormal Subgroup

In group theory, the study of subgroup structures provides essential insight into the behaviour and symmetry of algebraic systems. Among the more specialised concepts is that of the abnormal subgroup, a notion that captures an extreme form of non-normality within a group. Abnormality is closely related to conjugation, self-normalisation, and a hierarchy of other technical subgroup properties. This article outlines the definition, context, and major implications of abnormal subgroups within the broader framework of abstract algebra.

Definition and Core Properties

An abnormal subgroup HHH of a group GGG is defined by an especially strong conjugacy condition: for every element x∈Gx \in Gx∈G, the element xxx must lie in the subgroup generated by HHH together with its conjugate Hx=xHx−1H^{x} = xHx^{-1}Hx=xHx−1. Thus the combined structure ⟨H,Hx⟩\langle H, H^{x} \rangle⟨H,Hx⟩ is sufficiently large to contain the conjugating element itself. This requirement situates abnormal subgroups at the opposite extreme from normal subgroups, for which conjugation leaves the subgroup invariant.
One immediate consequence is that abnormal subgroups behave restrictively within the ambient group. Their conjugates interact so thoroughly with the group’s internal structure that they cannot be confined or controlled in the manner typical of normal or subnormal subgroups. This makes them particularly important in understanding groups with sparse normal structure or in analysing the limits of conjugacy-based subgroup behaviour.

Relationship to Normality and Self-Normalisation

A noteworthy implication of the definition is that every abnormal subgroup is self-normalising. The normaliser NG(H)N_{G}(H)NG​(H) of a subgroup HHH comprises elements of GGG that conjugate HHH into itself. However, if every element of GGG lies in the subgroup generated by HHH and its conjugate, any element outside HHH cannot preserve HHH under conjugation unless it already lies in HHH. Hence NG(H)=HN_{G}(H) = HNG​(H)=H.
This self-normalising property aligns abnormal subgroups with the notion of contranormality, describing subgroups whose normal closure is the whole group. In fact, every abnormal subgroup is contranormal, reflecting the extent to which conjugates of the subgroup collectively generate the entire group.
The relationship to normality is highly restrictive: the only subgroup that can be both abnormal and normal is the group itself. A normal subgroup that satisfied the abnormality condition would necessarily force every element of the group to lie in it, eliminating the possibility of proper abnormal normal subgroups.

Links with Weak Abnormality and Other Subgroup Conditions

Within the hierarchy of subgroup properties, abnormality stands at the top of a chain of increasingly weaker forms. One key implication is that every abnormal subgroup is weakly abnormal, a property characterised by relaxed versions of the generative requirements involving conjugates. Weak abnormality guarantees self-normalisation, so the implications cascade downward:

• abnormal ⟹ weakly abnormal ⟹ self-normalising.

These connections are particularly useful in classifying subgroups of finite groups or groups with significant symmetry constraints.
Abnormal subgroups also belong to several broader classes defined through conjugacy conditions. In particular, they are always pronormal. Pronormality requires that for any element x∈Gx \in Gx∈G, the subgroups HHH and HxH^xHx be conjugate within their join ⟨H,Hx⟩\langle H, H^x \rangle⟨H,Hx⟩. Abnormal subgroups satisfy an even stricter criterion, making pronormality a natural consequence. Pronormal subgroups, in turn, are weakly pronormal, connecting abnormal subgroups to a wider constellation of conjugacy-based subgroup behaviours.
Closely related is the classification of abnormal subgroups as paranormal. Paranormal subgroups satisfy the condition that any element x∈Gx \in Gx∈G conjugates the subgroup into a chain of subgroups ascending from HHH to HxH^xHx, ensuring that conjugation does not move the subgroup too far within the group structure. Abnormal subgroups automatically satisfy this condition because their generative closure with their conjugates is so comprehensive.
Furthermore, abnormal subgroups are necessarily polynormal, meaning that conjugation by successive elements of the group remains confined within a finite series of subgroups terminating at a conjugate of the original. This forms another layer of containment linking abnormal subgroups to larger families of conjugacy-controlled subgroup types.

Algebraic Significance and Applications

While abnormal subgroups are comparatively rare in many classical groups, they play an important role in the structural analysis of groups with complex or poorly behaved normal structures. Their presence can indicate that the group lacks intermediate subgroups with stable conjugacy behaviour, or that certain normal series cannot be refined. They arise naturally in solvable groups, in some infinite groups, and in contexts where the interplay between conjugation and subgroup generation is especially rigid.
Abnormality also serves as a limiting case that helps clarify the boundaries between different subgroup properties. By providing examples of extreme conjugacy sensitivity, abnormal subgroups help delineate the range of behaviour expected from pronormal, paranomal, or self-normalising subgroups. Within group-theoretic proofs, these properties may be invoked to restrict the behaviour of potential counterexamples or to demonstrate that certain subgroups cannot possess intermediate normalising features.
Although the concept is highly abstract, it contributes to a deeper understanding of how subgroup structures govern the internal symmetry of algebraic systems. Study of abnormality also intertwines with investigations into automorphisms, lattice structures of subgroups, and the behaviour of group actions on cosets.