Nonmormal

Nonmormal is a term used in group theory to describe a subgroup that is not normal in a given group. A subgroup H of a group G is nonmormal in G if there exists an element g in G such that gHg^{-1} ≠ H. Equivalently, H is not invariant under conjugation by all elements of G.

For a nonmormal H, the conjugates gHg^{-1} for g in G form a family of subgroups of the same order. The intersection of all these conjugates, called the core of H in G, is a normal subgroup of G contained in H and represents the largest normal subgroup of G contained in H. The normalizer N_G(H) = {g in G | gHg^{-1} = H} plays a key role: H is normal in N_G(H), and the number of distinct conjugates of H in G equals the index [G : N_G(H)]. The normal closure of H in G is the smallest normal subgroup of G containing H; it is generated by all conjugates of H.

Examples help illustrate the concept. In the symmetric group S3, the subgroup H = ⟨(12)⟩ is nonmormal in S3, since its conjugates are ⟨(12)⟩, ⟨(13)⟩, and ⟨(23)⟩. The core of H in S3 is trivial, and the normal closure of H in S3 is the whole group S3.

Nonmormal subgroups occur in many groups and are fundamental for understanding how a group acts by conjugation on its subgroups. They illuminate the relationship between a subgroup and the global structure of the ambient group through conjugacy, normalizers, cores, and closures.

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